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Computers and Chemical Engineering 33 (2009) 699–711

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

n the application of model reduction to plantwide control

ogdan Dorneanua, Costin Sorin Bildeab,∗, Johan Grievinka

Delft University of Technology, DelftChemTech, Julianalaan 136, 2628BL Delft, The NetherlandsUniversity POLITEHNICA of Bucharest, Department of Chemical Engineering, Str. Gh. Polizu 1, RO-011061 Bucharest, Romania

r t i c l e i n f o

rticle history:eceived 21 March 2008eceived in revised form9 September 2008

a b s t r a c t

This paper proposes an approach for obtaining reduced-order models of chemical processes with appli-cation to the design of plantwide control systems. The approach is based on the inherent structure thatexists in a chemical plant and identifies units or groups of units which determine the steady-state anddynamic behaviour of the plant. Specific reduction techniques with different accuracy are applied to each

ccepted 9 October 2008vailable online 22 October 2008

eywords:odel reduction

lantwide control

unit, followed by the coupling of the reduced models, according to the process flowsheet structure. Bymeans of a case study, the assessment of plantwide control structures for the iso-butane–butene alkyla-tion plant, the effectiveness of the approach is proven. Moreover, it is shown that the approach is ableto retain the nonlinearity of the original plant model and preserves the significance of important modelvariables.

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. Introduction

In today’s competitive environment, high economical per-ormance of chemical and biochemical plants is achieved byost-effective steady-state integrated design and by continuouslyesponding to the market conditions through dynamic operation.he desired policy of operation is accomplished by control systemshat maintain the steady-state or implement the optimal dynamicehaviour (Engell, 2007).

The synthesis, design and optimization of the plantwide controlystem require dynamic models of the chemical/biochemical plant.ften, the models are very complex, and therefore call for highomputational speed and high memory storage capacity. Moreover,he problems are likely to be ill-conditioned, since the applicationsnvolve solving an inverse problem. This often results in significantumerical troubles, unstable convergence and eventually, unreli-ble results.

Because of numerical and time constraints it is not always possi-le to consider models of the highest complexity. The model muste adapted to the specific purpose (Baldea, Daoutidis, & Kumar,006), although in many cases it is not evident which simpli-

cations are suitable. Therefore, for particular simulations, it isecessary to find an appropriate technique to reduce these effects.o achieve these objectives, the model quality is crucial. The modelhould predict with good accuracy the behaviour of the real system

∗ Corresponding author.E-mail address: s bildea@upb.ro (C.S. Bildea).

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098-1354/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2008.10.007

© 2008 Elsevier Ltd. All rights reserved.

t describes. Moreover, some process models which are developedrom experimental observations are far from being highly accurate.or this reason, it is important that the reduced model takes intoccount this uncertainty (Nagy & Braatz, 2003, 2007). In the sameime, the problems that need to be solved often imply repeatedolution of the process model. Thus, it is important that the com-lexity of the model is limited, in order to allow solution during aestricted time. Another requirement for the process models is theimited cost of maintenance which implies that the effort to adapthe model to future plant changes must be kept as small as possible.dditionally, in many applications of the chemical industry, the pro-ess is highly nonlinear and it is important that this characteristics preserved (Nagy, Mahn, Franke, & Allgöwer, 2007).

Model reduction is one technique that can be used for obtaininguch a model (Marquardt, 2001; Antoulas, Sorensen, & Gugercin,000). In this paper, a new approach to model reduction, with appli-ability to the dynamic models of chemical processes, is presented.his approach makes use of the inherent flowsheet structure thatxists in a chemical plant in the form of units or groups of units thatre connected by material streams. The decomposition mirrors theecentralization of the control problem. The recommended pro-edure exploits the knowledge about the process and applies theodel reduction techniques to individual units of the plant, and

hen couples together these reduced models.

The procedure is illustrated by means of a case study: the iso-

utane alkylation plant. We will demonstrate that an importantecrease of the simulation time is obtained while the reducedodel preserves the behaviour of the original model. Moreover,

aving reduced models of the units connected accordingly to

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rocess structure is a natural way of simplifying the adaptation touture plant changes.

The paper is structured in the following way: Section 2 exploresxisting model reduction techniques and the application of modeleduction to chemical plants. Next, the proposed approach to modeleduction is introduced in Section 3. The algorithm is applied tohe iso-butane alkylation plant in Section 4. Finally, conclusions areresented in Section 5.

. Model reduction

The models of dynamical systems contain a large number ofquations, both differential and algebraic, and, quite often, thesequations contain complex functional expressions. Such modelsan be written in the following, general form:

dx(t)dt

= f (x(t), u(t), ˛)

y(t) = g(x(t), u(t), ˛)(1)

ere t is the time variable, x(t) ∈Rn is the state vector, u(t) ∈Rm thenput vector, y(t) ∈Rp the output variable vector, ˛ is the parametersector, and n is the state space dimension. The dimension of thenput vector m and the dimension of the output variable vector p,re much smaller than n, and usually m ≥ p. The function f describeshe dynamics of the system, while the function g describes the wayn which the observations are deduced from the state and the input.

Model reduction has a long history in the systems and controliterature. The main idea of this technique is to reduce the com-lexity of the dynamic model, while preserving its input–outputehaviour. We recommend the review on model reduction tech-iques in Marquardt (2001). One can find several ways of classifyinghe model reduction methods, but probably the best way is the oneresented in the above-cited paper:

Model simplification—a technique that preserves the number ofequations of the original model, but reduces the complexity ofthe functional expressions in the model equations.Model-order reduction—a technique that replaces the originallarge-scale model with a model having (much) less equations.

.1. Model simplification

The model simplification techniques are very popular in the areaf chemical engineering. In many applications (for example inynamic optimization), the repeated evaluation of the right-handide of the model described by Eq. (1) is often the main computa-ional effort. Hence, by reducing the complexity of the functions toe evaluated, the model simplification methods are as importants the ones which reduce the number of equations.

Among these techniques, one can distinguish the linearizationround a nominal operating point, the approximation of functionalxpressions and the simplification of chemical kinetics (Nafe &aas, 2002) and/or physical properties models. In addition, sig-

ificant model reduction is also possible by model lumping (Ranzi,ente, Goldaniga, Bozzano, & Faravelli, 2001), in which thermo-ynamic phases and chemical components are taken together

nto pseudo-phases and lumped species. However, such lumpingan increase the complexity of the functional expressions in theemaining model equations.

.1.1. LinearizationThe linearization around a nominal operating point is one of the

ost often used methods to simplify nonlinear models. The maindea of this procedure is to rewrite the system presented in Eq. (1)

cal Engineering 33 (2009) 699–711

nder the following form:

dx(t)dt

= A · x(t) + B · u(t)

y(t) = C · x(t) + D · u(t)(2)

here x(t) = x(t) − xnom(t), u(t) = u(t) − unom(t), y(t) =(t) − ynom(t).

A, B, C, D are the system matrices and A ∈Rn×n, B ∈Rn×m, C ∈Rp×n

nd D ∈Rp×m, while the subscript nom represents the nominal oper-ting point.

One way of realizing the transformation is by using the Tayloreries expansion. The linearization reduces drastically the compu-ational effort. The simulation is more reliable since the nonlinearystem it is replaced by a linear system, easier to solve. However, theinear models do not adequately predict the process dynamics forhe cases when the system is far away from the nominal operatingoint.

.1.2. Approximation of functional expressionsThis technique implies the replacement of a complicated, non-

inear model equation by simpler functional expressions. Thisimple functional expression has to approximate the original up touser specified tolerance, which leads to a multi-variate nonlinearpproximation problem.

.1.3. Simplification of physical properties modelThe most common method of simplifying the physical proper-

ies models is the development of local thermodynamic models.he use of local thermodynamics reduces significantly the com-utational effort, while maintaining the accuracy at an acceptable

evel.The easiest way of reducing the complexity is to re-write the

unctional expression of a property in a polynomial form.

.1.4. Simplification of chemical kineticsIn many applications, large-scale chemical kinetic models are

uite common. The complexity arises from the large number ofeactions and components, and it can be reduced both by eliminat-ng from the reaction network: (a) reactions (Bhattacharjee, Schwer,arton, & Green, 2003), which is a task of the model simplification,nd (b) species, which is a special case of model-order reduction.he use of moments in population balance models, for eliminatinghe distributed nature of the population balance, is an example ofhis technique.

.2. Model-order reduction

Reducing the complexity of the functional expressions in theodel equations is not always enough to achieve the model reduc-

ion. In many cases, the number of the model equations needs to beeduced also. In the model-order reduction class, two main groupsf methods can be distinguished: linear and nonlinear model-ordereduction, depending on the type of the full model.

.2.1. Linear model-order reductionThe largest group of order-reduction methods applies to the

inear systems.The problem can be stated in the following form:

Given the linear dynamical system (2), one should find an approx-

imation

dx(t)dt

= A · x(t) + B · u(t)

y(t) = C · x(t) + D · u(t)(3)

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B. Dorneanu et al. / Computers and C

with A ∈Rk×k, B ∈Rk×m, C ∈Rp×k, D ∈Rp×m and k � n such that thefollowing properties are satisfied (Antoulas et al., 2000):• The approximation error is small, and a global error bound can

be defined.• System properties, like stability, are preserved.• The procedure is computationally stable and efficient.

The approach has many advantages. First, the linear models areasy to be obtained, for example through Taylor series expansion.n the same time there is a considerable large range of “treatment”echniques for linear models, which spreads from the backwarduler method to multistep methods of solving the ordinary differ-ntial equations (ODEs) that describe the system. And, last but noteast, stable and well-understood numerical linear algebra algo-ithms allow the analysis, control or simulation of such a system.

Many methods of obtaining the approximation presented in Eq.3) have been developed in different fields, including control, fluidynamics, and chemical engineering. They can be classified intowo categories: projection-based methods and non-projection basedethods.

The common feature of the projection-based methods is thathey are trying to find such k and n–k dimensional subspaces S1nd S2 of the state space, S, that the reduced system will resultrom the projection of the state onto S1 and the residual onto S2, andhe elimination of the states in S2. In this category one can mentionechniques such as Krylov-subspace or momentum matching meth-ds, balanced realization-based methods or proper orthogonalecomposition (POD)-based methods (Skogestad & Postlethwaite,996; Antoulas & Sorensen, 2001; Rathinam & Petzold, 2003; Penzl,006).

In the non-projection methods, the state space of the approx-mation has no connection with the unreduced system. The

ost used techniques in this category are: Hankel optimalodel reduction method, singular perturbation method or various

ptimization-based methods (Mäkilä, 1991; Marquardt, 2001).Each of the methods above has advantages and drawbacks.

or example, the Krylov-subspace-based methods can be appliedor a very high-order system, but have robustness, stability andfficiency issues (Bai, 2002). In the case of the Hankel-normpproximation and balanced realization methods, very low-rankpproximations are possible and accurate low-order models willesult, but the solution requires dense computations and can bearried out only for low-dimension (few hundreds equations) mod-ls (Antoulas & Sorensen, 2001). In the same time, the accuracyf the result is guaranteed by theoretical results and can be cal-ulated a priori (Skogestad & Postlethwaite, 1996). The POD-basedethods are not system invariant, the resulting simplification being

eavily dependent on the initial excitation, but can be applied forigh-complexity systems (Antoulas & Sorensen, 2001).

.2.2. Nonlinear model-order reductionMost nonlinear model-order reduction methods are extensions

f linear methods for nonlinear systems (Scherpen, 1994). Forxample, some reduction techniques apply linear projection to theonlinear model as if it were a linear model. The simplest approach

or generating reduced-order models for nonlinear systems is basedn linearization of systems nonlinearity and subsequent applica-ion of linear model-order reduction methods. The main drawbackf this approach is that the obtained reduced model is valid onlyocally, around the initial operating point of the nonlinear system.

ence, these techniques do not work for cases when the mod-ls have a strongly nonlinear behaviour. Since these approachesre not always successful, due to the high nonlinearity of thehemical and biochemical plant models, some particular meth-ds have been developed, for example the use of neural networks

cal Engineering 33 (2009) 699–711 701

Prasad & Bequette, 2003), or the use of hybrid models (Nagy etl., 2007). However, the main issue in working with nonlinearodel-order reduction is that the reduced models obtained can

e even more difficult to solve than the full models (Marquardt,001).

.3. Linear model-order reduction by balanced realization

The linear model-order reduction methods are easy to apply andhe accuracy of the obtained reduced models is guaranteed by the-retical results. The linear model is easy to obtain, sometimes onlyy means of Taylor series expansion. In the same time, the models easy to solve and analyze, by means of stable, well-understoodumerical linear algebra algorithms.

In the following we will refer to methods for obtaining reduced-rder models by balanced realization. In this approach, a new statepace description is obtained in such a way that the controllabilitynd observability gramians are equal and diagonal. The balancedealization can be easily obtained by simple state similarity trans-ormations, and routines for doing this are available in commercialoftware, like MATLAB®.

The most important property of a balanced realization is thatfter the balancing, each state is as controllable as it is observable,nd the measure of state’s observability and controllability is giveny its associated Hankel singular value (Skogestad & Postlethwaite,996). This property becomes interesting for the model reductionechniques, which remove the states with little effect on the sys-em’s input–output behaviour.

Considering the balanced linear model, which can be described

y Eq. (2), the state space vector x is partitioned into

[x1x2

], where

˜2 is the vector of the states to be removed. After partitioning theatrices A, B and C, Eq. (2) is rewritten as follows:

dx1(t)dt

= A11 · x1(t) + A12 · x2(t) + B1 · u(t)dx2(t)

dt= A21 · x1(t) + A22 · x2(t) + B2 · u(t)

y(t) = C1 · x1(t) + C2 · x2(t) + D · u(t)

(4)

Two main approaches to reduce the order of the model by bal-nced realization can be mentioned:

(a) Balanced truncation—removes all the states corresponding tothe small Hankel singular values. By eliminating the states withlittle effect on the system’s input–output behaviour (x2), Eq. (4)become Eq. (3), with x(t) ≡ x1(t), A ≡ A11, B ≡ B1, C ≡ C1 andD ≡ D. An important property of the truncation is that it retainsthe system behaviour at high frequency.

b) Balanced residualization—sets to zero the derivatives of thestates with little effect on the system’s input–output behaviour.An important property of the residualization is that it preservesthe steady-state gain of the system. In this case, the matrices inEq. (3) are calculated as follows:

A ≡ A11 − A12 · A−122 · A21

B ≡ B1 − A12 · A−122 · B2

C ≡ C1 − C2 · A−122 · A21

D ≡ D − C2 · A−122 · B2

(5)

the balanced realization techniques for obtaining the reducedmodels is that the error can be calculated a priori. The infinitenorm of the error between the original model and the reduced-order model is twice the sum of the Hankel singular values ofthe eliminated states (Skogestad & Postlethwaite, 1996).

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. Model reduction with process knowledge

As seen in the previous section, many reduction techniques haveeen developed, in order to obtain models that:

predict the behaviour of the plant with good accuracy,have limited complexity, to allow repeated solution during arestricted time.

Although significant reduction of the number of equations cane achieved, the benefit is often limited, because the structure of theroblem is destroyed, the physical meaning of the model variables

s lost and there is little or no decrease in the solution time (Van denergh, 2005). Moreover, especially in the case of nonlinear model-rder reduction, the obtained reduced model is often not easier toolve.

Another desired quality of the reduced model is the adaptabilityo future plant changes. If in an industrial application, one of thenits in the plant flowsheet is replaced, a new reduced model ofhe plant has to be obtained.

.1. Classical model reduction algorithm for chemical plants

In the following sections, the focus will be on obtaining reduced-rder models that can be used to determine the plantwide controltructure of the plant. The main steps of a classical approach toodel reduction will be presented, together with the most common

ssues that appear when applying the procedure.

.1.1. Obtaining the rigorous modelThe starting point of the approach is a rigorous dynamic model of

he plant. This dynamic model is obtained using a commercial pack-ge, such as Aspen Dynamics® or gPROMS®. Usually, a basic controlf inventory at unit level is included. In the same time, depend-ng on the package that is used, some model reduction is alreadyresent, such as the use of flow-driven simulation instead of theigorous pressure-driven mode, the local thermodynamic models,he simplified chemical kinetics or the instantaneous models for

ixers, pumps, valves or heat exchangers. Very often, the user isble to insert own equations and data to describe parts of the plant.nder these conditions, the plant model still contains thousands ofifferential and algebraic equations (DAEs), with initial conditionserived from a nominal steady-state operating point. The size of theodel can lead to numerical difficulties when trying to find a solu-

ion. When this happens, the search for the cause of the difficultiesnd for a remedy is a time-demanding task. Solving the dynamicodel implies solving numerically a system of DAEs. The numerical

lgorithms often require an initial guess for the algebraic variables.epending on the problem, finding the best starting point for thelgorithms can also be a challenging task.

.1.2. Obtaining the linear modelThe linear model in state space formulation is obtained easily

ith the help of the dynamic simulators discussed above. Deter-ining the stability of the model is also an easy task. All it takes

s to calculate the eigenvalues of the system matrix, obtained afterhe linearization. However, for unstable systems, the origin of thenstability is often very difficult to identify.

We stress that many chemical and biochemical plants have atrongly nonlinear behaviour. Some of the most common features

re the high parametric sensitivity and the state multiplicity. Theseffects are enhanced by coupling the units through heat integrationr material recycles. For this reason, the linear models are reliableear the linearization point, but their accuracy is poor for largeisturbances.

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cal Engineering 33 (2009) 699–711

.1.3. Obtaining the balanced state space modelThe balanced realization methods for model reduction are now

well-developed and relatively simple group of techniques. For lin-ar systems, the approach requires only matrix computation. In theame time, the error bounds between the original and the reducedodel can be easily and a priori calculated with the help of the

nduced Hankel norm (Skogestad & Postlethwaite, 1996). For sta-le systems, the balancing of the state space is straightforward. Ifhe system is unstable, the stable part is isolated, balanced, anddded back to the unstable part of the model. For very large models,he algorithm often fails due to the ill conditioning of the Lya-unov equation solved while calculating the gramians (MullhauptRiedel, 2004).

.1.4. Obtaining the reduced-order modelIn the next step, the reduced-order model is obtained from

he balanced space model. Reduction of the system is achievedy retaining only certain states in the representation. This oper-tion is equivalent to defining a certain subspace within the statepace. Small Hankel singular values in the balanced realization indi-ate state variables with little contribution to the input–outputehaviour. The reduced-order model is obtained by equating toero these variables (truncation) or their time-derivatives (resid-alization). Truncation is more accurate in representing the initialart of the dynamic response, but residualization preserves theteady-state gain. If the dynamics of the systems is high-order, noignificant reduction can be achieved.

.1.5. Plantwide control structuresThe term plantwide control refers to all the structural and strate-

ic decisions which have to be taken in order to accomplish aesired policy of operation. Two approaches are often used forhe assessment of the plantwide control structures (Skogestad &arsson, 1998):

(i) Several alternatives for the plantwide control structure are eval-uated, for a given design of the plant. The main disadvantageof this approach is the large number of alternatives to be gen-erated and assessed. In the same time, it is difficult to have aprecise problem definition. This can lead to several solutionsfor the same plantwide control problem (Larsson & Skogestad,2000).

ii) The plantwide control and the design are carried on in the sametime. This approach (Bansal, Perkins, Pistikopoulos, Ross, &van Schijndel, 2000; Schweiger & Floudas, 1997) involves, veryoften, solving multi-objective optimization problems, based onfull nonlinear dynamic model of the process.

The design of the plantwide control structures also brings somearticularities when model reduction is considered. Although theoal is a control structure for the whole plant, many control loopsre local to certain units. For example, instabilities arising from heatntegration are solved by manipulating local heat duties; in distilla-ion, composition of product streams is controlled by reflux rate oreboiler duty. From a plantwide viewpoint, the design of the con-rol structure is mainly concerned with the inventory of reactants,roducts, impurities and by-products (Bildea & Dimian, 2003). Theolution is much simpler by excluding the local control loops fromhe analysis. This can be done by including these local control loopsnto the reduced model of the unit.

Both approaches need, however, a dynamic model of the plant.n the same time, this model has to be solved many times until aolution to the plantwide control structure is reached. If the models solved fast, the control structure alternatives are easily evalu-ted.

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.2. Model reduction with process knowledge algorithm

After presenting the classical method to obtain the plantwideontrol structure of the plant using the model reduction andhe main difficulties that can be encountered, a new approachs further introduced. This approach uses the knowledge abouthe structure that exists in a chemical plant in the form ofnits that are connected by material streams, which mirrors theecentralization of the control problem (Vasbinder & Hoo, 2003).he recommended procedure is to apply model-order reductionnd/or model simplification to individual units, and then to cou-le these reduced models. The result of this coupling is a reducedodel of the full plant. The effectiveness of the approach will be

roven by means of a case study: the iso-butane–butene alkylationlant.

To obtain the reduced model of the plant, the following stepsave to be considered:

. Identification of units or groups of integrated units to which localcontrol is applied

There are several ways to split the plant flowsheet into units.For example, the flowsheet can be divided into units according tothe equipment type: mixers, pumps, reactors, heat exchangers,distillation columns, etc. For the distillation columns, an evenmore detailed decomposition can be considered: reboiler, trays,and condenser. Another idea is to split the flowsheet consideringthe main operations: mixing section, reaction section, separa-tion section, etc. Therefore, the number of individual blocks tobe further analyzed depends on the criteria used for flowsheetdecomposition. For the case of a very fine decomposition, a largenumber of blocks are obtained, which lead to a large increaseof the time allocated for individual block analysis. When thedecomposition is too raw and the number of blocks is too small,the attached value of the approach is insignificant.

One way of achieving the flowsheet decomposition is byusing the concept of controlled group of units (CGU, Naka, Lu,& Takiyama, 1997). The concept of CGU represents an inde-pendently operable part of the flowsheet, which provides aninventory control function so as to maintain it at a certainsteady-state for some time. Each CGU has an internal structurecomprised of a number of processing units and control loops.The decomposition into CGU’s should be done in such way thatthe dynamic interactions between them are minimized (Nakaet al., 1997, Seki & Naka, 2006). To achieve this, the connectionbetween the parts of the plant through mass and energy streamsare considered. Then decomposition is done by “cutting” streamsof material, while units connected by an energy stream are kepttogether.

. Application of tailored reduction techniques to individual unitsor groups of integrated units

Each of the models of the units has smaller size compared tothe original full model. Therefore, simple methods such as thereduction of the balanced state space model (which are not suit-able for large scale models) are easy to apply and to implementnumerically for the purpose of the model reduction. Moreover,since the error bounds between the original and the reduced-order model can be calculated a priori (see above), the cut-offcriteria for the smaller Hankel singular values for different pro-cess units is defined in a consistent way. For the units with astrongly nonlinear behaviour, the use of model simplification or

nonlinear model-order techniques can be a very good solution.In addition, due to the smaller size of the unit model, the chancesof obtaining a reduced model which is difficult to solve decreasesignificantly.

. Obtaining the reduced model of the full plant

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cal Engineering 33 (2009) 699–711 703

The reduced model of the full plant is obtained by connectingthe reduced models of the individual units. The connection isrealized through streams characterized by the flowrates of eachchemical species, temperature and pressure. To assess the qual-ity of the reduced-order model, various disturbances are appliedon one of the feed streams of the unit or plant. The response inthe outlet stream is recorded and compared further with theresponse predicted by the full-order model, for the same distur-bances.

Fig. 1 introduces the algorithm of model reduction with processnowledge. The steps that need to be performed in order to obtainhe reduced model are

1) A rigorous dynamic (nonlinear) model of the plant is developed.This step is performed using several commercial packages, asmentioned in Section 3.1.

2) Identify the CGU’s in the plant flowsheet. The concept of CGU’sis presented above.

3) Achieve the decomposition of the plant flowsheet. This decom-position should “cut” only material streams.

4) The rigorous dynamic model of each CGU is developed usingcommercial packages. At this point, control structures for theCGU are added, if necessary.

5) Proceed to the model reduction of each CGU. The following sub-steps have to be performed:a. Obtain a linear model of the CGU. As mentioned above, many

commercial packages can perform this task easily.b. The balanced realization of the linear model of the CGU is

obtained.c. The model reduction of the balanced model is performed.

6) Check if the resulting CGU’s reduced models are accurateenough. This can be achieved by comparing the behaviour pre-dicted by the reduced and the rigorous models of each CGU,either in the time domain or in the frequency domain (or both).The comparison can be graphical or statistical. If the result-ing reduced model is not accurate enough, another techniqueshould be used.

7) If the reduced models of the CGU’s are accurate, the full modelof the plant is reconstructed by coupling together the reducedmodel of the CGU’s.

8) Check if the obtained full model of the plant is accurate. This isdone by validation against the original nonlinear model. If thefull model is not accurate enough, return to step (2). Otherwise,the algorithm stops.

In the following section, the effectiveness of the approach wille proven by means of a case study.

. Case study: plantwide control of iso-butane–butenelkylation plant

The alkylation of iso-butane with butene is a widely usedethod for producing high-octane blending component for gaso-

ine. In industrial practice, the modern processes use the sulphuriccid as a catalyst. Although the chemistry of alkylation is very com-lex, for our purposes, the following reactions will be used for theverall chemistry:

4H8(A)

+ i − C4H10(B)

→ i − C8H18(P)

(6)

4H8(A)

+ i − C8H18(P)

→ C12H26(R)

(7)

This particular case study has been chosen because the planttructure includes the most common units that can be seen in

704 B. Dorneanu et al. / Computers and Chemical Engineering 33 (2009) 699–711

Fig. 1. The algorithm of the model reduction with process knowledge.

B. Dorneanu et al. / Computers and Chemical Engineering 33 (2009) 699–711 705

Fig. 2. The basic iso-butane alkylation plant flow scheme.

Table 1Distillation column design specifications.

Column specifications Units Column 1 (COL1) Column 2 (COL2) Column 3 (COL3)

Number of stages – 28 29 24Feed stage – 18 10 11Reflux ratio – 110 0.1 0.2DRC

tebt(

4

istillate to feed ratio – 0.017eboiler duty MW 6.65ondenser duty MW −6.28

he industry: heat integrated reactor, distillation columns, mix-

rs, pumps, recycles. The structure of the plant flowsheet wille presented in the following, while a detailed description ofhe process can be found in the book of Dimian and Bildea2008).

8t

Fig. 3. The iso-butane–butene a

0.95 0.913.8 0.09

−3.54 −0.33

.1. Process description and the rigorous model

The plant produces 26 kton/year iso-octane, with a selectivity of5% for a yearly run of 8000 h. The raw materials are the butene andhe iso-butane. The butene stream has a large amount of propane,

lkylation plant flowsheet.

706 B. Dorneanu et al. / Computers and Chemical Engineering 33 (2009) 699–711

Table 2Stream summary for the iso-butane alkylation plant.

Stream name (see Fig. 3) Temperature ( ◦C) Pressure (bar) Mass flow (kg/s)

A B P R I

FB0 25.0 8.0 – 0.580 – – –FA0 + FI0 25.0 8.0 0.550 – – – 0.108FIN 2.3 8.0 1.378 10.662 0.001 Trace 0.110FOUT −5.0 8.0 10.409 126.461 11.564 1.734 1.379F8 48.9 9.0 0.833 10.117 0.925 0.139 0.110INERT 26.8 8.0 0.001 0.054 Trace Trace 0.108F9 61.1 9.0 0.832 10.063 0.925 0.139 0.002REC 57.0 8.0 0.828 10.054 0.001 Trace 0.002FIH

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wIbutane alkylation plant using this approach. For the assessmentof the plantwide control structure other methods of obtaining thereduced model have to be used. The following steps will make useof the knowledge about the process while achieving the modelreduction.

10 194.3 8.3-OCTANE 93.5 1.2EAVIES 228.2 1.5

hich is an inert for the reaction, as impurities (10%). The mainroduct is the iso-octane. Since the selectivity of the process is not00%, some heavy products (modelled as dodecane) are formed.

The process consists of two basic operations (Fig. 2). The mainperation is the alkylation of iso-butane with butene to iso-octanend dodecane which occurs in the reaction section. The two reac-ants are converted into products at low temperature, in a reactor

odelled as a CSTR. The reaction volume is 4 m3 and the reactionressure is 8 bar. In practice, the temperature ranges between 0nd −10 ◦C. In this study, the reaction temperature is chosen to be5 ◦C. The cooling is achieved in an external heat exchanger. Foreat integration purposes, the reactor outlet stream acts as coolanto the reactor inlet stream, as seen in Fig. 3.

After reaction, the main product is separated from the reactorutlet stream. The separation is done by distillation, through theirect sequence. The inert is removed in the first column, whilehe reactants, having similar volatilities, are removed next. In thisay only one recycle stream appears in the flowsheet. The third

olumn separates the product from the heavies. A summary of theistillation columns design specifications is presented in Table 1.

Once the plant structure has been decided, the next step is thequipment design and sizing. This was achieved using Aspen Plus®.summary of the main streams composition is presented in Table 2.

After obtaining a steady-state model in Aspen Plus®, a dynamicodel of the iso-butane alkylation plant is developed in Aspenynamics®. Aspen Dynamics® creates a very basic plantwide con-

rol structure consisting of a push flow control scheme on the rawaterial feeds, pressure control and where required, level control.owever, this basic control structure does not involve any qualityontrol loops. We will call this basic dynamic model the “open-oop” model and we will analyze it further.

The simulation of the “open-loop” plant shows that the plant isnstable. Starting from the nominal (steady-) state, the reactor tem-erature drops fast and all flow rates dramatically increase (Fig. 4).oon, one level control loop reaches its limits, and overflow occurs.

.2. The linear model

The original plantwide model is linearized around the nominalperating point (steady-state). This step is done using the lineariza-ion toolbox provided by Aspen Dynamics®. The system matrix Aas three positive eigenvalues, with unclear origin: the instabilityould be determined by the heat integrated reactor, which presentsstrongly nonlinear behaviour, as it will be shown later; another

ource of instability could be the reactor-separation-recycle struc-ure; it is also possible that one or more distillation columnsresent multiple steady-states, phenomenon which is accompa-ied by instability (Jacobsen & Skogestad, 1995). An experiencedodeller will consider other possible causes of the plant’s instabil-

Ft

0.004 0.008 0.924 0.139 Trace0.004 0.008 0.923 0.00 TraceTrace Trace 0.001 0.139 Trace

ty. The issue of the three positive eigenvalues needs to be analyzedurther.

.3. The balanced model

Obtaining the balanced model using MATLAB® fails: thelgorithm complains about ill-conditioned matrices. A balancedealization of the linear model of the original plant, which has 554tates, cannot be obtained. The cause is not easy to be determined.

As seen above, the use of reduced-order models, in a classicalay, for designing the plantwide control seems to be hopeless.

t looks impossible to develop a reduced model for the iso-

ig. 4. Instability of the “open-loop” plant: (a) the temperature of the reactor; (b)he mass flow of the column 1 (COL1) bottom stream.

hemical Engineering 33 (2009) 699–711 707

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B. Dorneanu et al. / Computers and C

.4. Exploiting the structure

For the purpose of obtaining the plantwide control structure,he flowsheet of the iso-butane alkylation plant is decomposed intonits and groups of units to which local control is applied. Theecomposition does not cut the connection between units that areeat integrated.

The plant model is split into the following units: mixers, reactortogether with the heat integration around it), distillation columns,nd pumps (Fig. 3). The mixers and the pumps are considered to benstantaneous, so they are not interesting from the point of view ofhe model reduction.

For the models of each unit, the molar flowrates of each com-onent, the temperature and the pressure for the inlet stream areonsidered as inputs, while the molar flowrates of each component,he temperature and the pressure of the outlet stream/streams areonsidered as outputs.

The model reduction techniques will be applied further to theeactor and each distillation column, individually.

.4.1. The reactor subsystemThe analysis of the reactor subsystem shows instability. How-

ver, the unit can be easily stabilized by adding a temperatureontroller manipulating the cooling duty. After the stabilization, thenit is linearized around the steady-state point. The linear modelhich is obtained has 15 states, and the agreement with the non-

inear model is good, as seen in Fig. 5a.Further, the reduction of the linear balanced model is attempted.

he reduced-order model has only seven states, but the agreementith the full nonlinear model is questionable (Fig. 5a). It can be

oncluded that, for this case, no significant order-reduction at highccuracy can be achieved.

ig. 5. Model reduction of the reactor sub-system: iso-octane molar flow in theutlet stream: (a) time response of the nonlinear, linear and reduced model for atep of 1 kmol/h on the butene in the feed; (b) time response of the nonlinear andimplified model for a step of 5 kmol/h on the butene in the feed.

w

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n

4

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Fig. 6. The instability of column 1 (COL1): distillate stream molar flow.

A second approach is considered in order to reduce the size ofhe reactor system model, preserving in the same time its non-inearity. This reduction is based on model simplification. A newynamic model is written, consisting of five component balances,nd considering constant temperature and physical properties:

dnA

dt= F (IN)

A − F (OUT)A − V · r1 − V · r2

dnB

dt= F (IN)

B − F (OUT)B − V · r1

dnP

dt= F (IN)

P − F (OUT)P + V · r1 − V · r2

dnR

dt= F (IN)

R − F (OUT)R + V · r2

dnI

dt= F (IN)

I − F (OUT)I

(8)

here F (IN)i

= the molar flow of component i in stream FIN (kmol/h);(OUT)i

= the molar flow of component i in stream FOUT (kmol/h);i = the number of moles of component i (kmol); i = A, B, P, R, I;= the reactor volume (m3); rj = the rate of reaction j (kmol/m3 h).

A constant temperature inside the reactor is achieved in prac-ice by the use of a temperature controller. In this way, there is noeed for the energy balance equation and one differential variableecomes an algebraic one. Moreover, the equations for determininghe temperature-dependent physical properties of the componentsre no longer necessary.

The control of the heat integrated reactor consists of a tem-erature controller, manipulating the cooling duty, and a levelontroller, manipulating the outlet flow.

As seen in Fig. 5b, the agreement with the AspenDynamics®

onlinear model is excellent.

.4.2. The distillation columnsUnexpectedly, the first and the third distillation column are also

nstable. It should be remarked that the dynamic model specifies,ealistically, the reflux on a mass basis, in contrast to the steady-tate simulation which uses moles. Realizing the large relativeolatilities and the very different molar weights, the existence ofultiple or unstable steady-states becomes a possibility (JacobsenSkogestad, 1995). Indeed, switching to mole-based specifica-

ions stabilizes the columns (Fig. 6). However, in the followinghe distillation columns will be modelled with mass-based refluxpecification and temperature controllers will be used to obtain

closed-loop stable system. Therefore, the control of the three

istillation columns consists of temperature control both in theop and in the bottom of the columns, in addition to the inven-ory control loops (pressure, levels in the reflux drum and columnump).

708 B. Dorneanu et al. / Computers and Chemi

Fig. 7. Model reduction for column 1 (COL1): iso-octane molar flow in the bottomsmf(

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tream: (a) time response for the full nonlinear and reduced model for a step ofagnitude 5 kmol/h (∼10%) on the feed stream butene molar flow; (b) time response

or the full nonlinear and the small order model for a step of magnitude 5 kmol/h∼10%) on the feed stream butene molar flow.

Similarly to the reactor system, the model of each distillationolumn is linearized and then the reduced-order model of the bal-nced realization is obtained.

For the distillation columns, a significant order-reduction ischieved with excellent accuracy, as seen in Fig. 7a, where resultsre presented for the first distillation column. From a nonlinearodel with about 200 states, the reduced-order model is linear and

as around 20 states. A summary of the model reduction results isresented in Table 3.

However, from the appearance of the time response of the fullonlinear model (Fig. 7a), it should be remarked that the behaviouran be well approximated by a model of even lower order, such assecond-order plus dead-time model with parameters estimatedith the help of the system identification toolbox provided byATLAB®.Fig. 7b compares the predictions of the nonlinear and the

econd-order and dead-time models of the first distillation column,onsidering the iso-octane molar flowrate in the bottom stream as

utput. The agreement between the two models is excellent.

For the further steps in assessing the plantwide control struc-ures the order-reduced models obtained from the balancedealization of the linear models will be used.

able 3odel reduction results.

nit Model reduction technique Full nonlinear model Reduced model

STR Model simplification 15 states 5 states

OL1Model-order reduction

185 states 20 statesOL2 191 states 25 statesOL3 163 states 22 states

imulation time 150 s 30 s

slToeari

tsi

lw

cal Engineering 33 (2009) 699–711

.5. Assessment of the plantwide control structure

At this point, three different models are available:

a) the original nonlinear model (R),b) the reduced-order linear model (M1),c) the reduced-order nonlinear model (M2).

The original nonlinear model, R, is developed in Aspen-ynamics® and will be taken as reference. Model M1 contains the

educed-order linear model of each unit in the flowsheet, whileodel M2 contains the reduced-order linear models of the distil-

ation columns and the simplified, nonlinear model of the reactor.Different plantwide control structures are considered and eval-

ated. In the following, we will discuss two of them (Fig. 8).In control structure CS1, the fresh feeds of both reactants are

xed. This has the advantage of setting directly the production ratend the product distribution. The disadvantage of this control struc-ure is that is highly sensitive to disturbances, as it will be shownater.

In control structure CS2, the butene fresh feed is fixed, but theso-butane is brought into the process on inventory control. Thisliminates the disadvantage of the previous control structure.

As the two control structures presented above differ only in theay the reactants are being brought into the process, the reduced-rder models of the reaction and separation section can be easilyeused.

In order to assess the quality of the reduced models M1 and2 and their applicability to plantwide control design, the control

tructures CS1 and CS2 are applied and the predictions of the fullnd reduced models are compared.

When the butene feed is decreased by 10%, the full model pre-icts that control structure CS1 is not able to control the plant: themount of butene in the process is not enough to consume the iso-ctane. The reactant is accumulating in the plant and the recycleow is continuously increasing. While simulating the full nonlin-ar model R, because of the high increase of the recycle flow, thespen Dynamics® simulation crashes and has to be stopped. This

s also the reason for the smaller amount of data available for thisodel. The nonlinear model M2 is correctly predicting the high

ensitivity to disturbances of this control structure, as shown inig. 9. However, the linear model M1 wrongly indicates that a newteady-state is reached. In other words, having a linear reduced-rder for the reactor does not preserve the system’s behaviour. Itan be concluded that the reduced-order model must be able toreserve the nonlinear behaviour of the full model.

When the same disturbance is applied to the control structureS2, a new steady-state is reached in relatively short time for theonlinear model R and M2, as seen in Fig. 10. The excellent accu-acy of the nonlinear reduced-order model, M2, is obvious. In theame time, an important reduction of the computation time, fromess than 3 min to about 30 s, should be also remarked (Table 3).his reduction of the computation time could result from the usef linear reduced-order models for the distillation columns. The lin-arization has as a result the reduction of the computational effort,s discussed above. In addition, the use of a simple model for theeactor, which consists only of five mass balance equations, has anmportant contribution to this reduction of the computation time.

From the results presented in Figs. 9 and 10 it can be concludedhat the control structure CS2 performs better than the control

tructure CS1 for the case presented above. This control structures recommended to be used further.

In the discussion above, we presented the way the rigorous, theinear and the reduced models were obtained. Since for this purpose

e used several tools, in the following we will summarize the way

B. Dorneanu et al. / Computers and Chemical Engineering 33 (2009) 699–711 709

res fo

toPs

Fr

Fig. 8. Control structu

hese tools were connected in order to achieve our goal. The rigor-us model was obtained using the Aspen Engineering Suite®. Aspenlus® and Aspen Dynamics® were used for obtaining the steady-tate and the dynamic model of the alkylation plant, respectively.

ig. 9. Dynamic simulation results, for 10% decrease of the fresh butene flowrate:ecycle flowrate.

TtsM

Fp

r the alkylation plant.

he linear models were determined with the help of the lineariza-ion toolbox in Aspen Dynamics®. The balanced realization of thetate space model and the model reduction were performed inATLAB®, as well as the parameter identification for the reduced

ig. 10. Dynamic simulation results, for 10% decrease of the fresh butene flowrate:roduction rate.

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10 B. Dorneanu et al. / Computers and

odels of the distillation columns. Finally, the simplified reactorodel and the complete reduced model of the alkylation plant were

btained using SIMULINK®.The connection between these tools was done through the

atrices of the state space representation of the system (the alkyla-ion plant). The output of Aspen Dynamics® linearization tool is theet of matrices presented in Eq. (2). With the help of these matri-es, the linear system is very easy to implement in MATLAB®, as atate space model. The model reduction performed in MATLAB®

o this state space returns another set of smaller size matrices,hich are also easy to turn into a state space model. SIMULINK®

s a dynamic simulation package which allows the user to specifyblock diagram representation of a dynamic process. As an exten-

ion to MATLAB®, the state space models obtained previously areully compatible.

Another point that needs to be discussed is the time needed forhe development of the reduced-order model using the knowledge-ased approach, assuming that a full nonlinear model is alreadyvailable. We estimate that an experienced modeller will develophe reduced-order model of the alkylation plant is not more than 2ays. However, obtaining a nonlinear reduced model of a unit cane more time consuming.

. Conclusions

The paper proposes and demonstrates the advantage of con-idering the structure of a process flowsheet when developingeduced-order models to be used during the design of thelantwide control system.

The recommended procedure is to apply the model reduction tohe individual units of the plant, and then to connect together theeduced models.

Some advantages of applying this procedure have been high-ighted by means of a case-study:

The procedure is flexible. Tailored model reduction techniquesare applied to different plant units. In the case presented above,the reactor model was reduced by model simplification, while lin-ear model-order reduction of the balanced realization was usedfor the distillation columns. The dynamics of the mixers, pumpsand the heat exchangers was neglected.The solution time is significantly reduced, by factor 5.The nonlinearity of the model can be preserved. This feature ishighly desirable since many units of chemical and biochemicalplants have a strongly nonlinear behaviour, such as state multi-plicity or parametric sensitivity, behaviour which is enhanced bythe coupling through heat integration and/or material recycles. Itwas shown that considering a linear model for the reactor in thereduced model of the plant did not preserve essential behaviouralfeatures of the system. After applying a disturbance for the controlstructure CS1, the linear reduced-order model did not predict thesystem sensitivity, in contrast to the predictions of the full-orderand the reduced-order nonlinear models.The maintenance and adaptation to future plant changes is facil-itated by the modularity of the reduced model. Once one of theunits in the plant flowsheet needs to be replaced, its reducedmodel can be easily inserted in the reduced model of the plant.The modularity of the reduced model is also useful for the casewhen a rigorous model for one of the plant units is not available.However, if a simplified model of the unit is available and accu-

rate enough, it can be easily inserted in the reduced model of thewhole plant.

After demonstrating the advantages of using this model reduc-ion technique for the assessment of the plantwide control

S

S

cal Engineering 33 (2009) 699–711

tructure of a chemical plant, the next step is to use it for otherpplications in the chemical industry. For example, dynamic opti-ization of plant operation is another time consuming task forhich the advantage of using reduced models is obvious.

cknowledgements

This project is carried out within the framework of MRTN-T-2004-512233 (PRISM-Towards Knowledge-Based Processingystems). The financial support of the European Commission isratefully acknowledged.

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