Adaptation strategies for real-time optimization

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Computers and Chemical Engineering 33 (2009) 1557–1567

Contents lists available at ScienceDirect

Computers and Chemical Engineering

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

daptation strategies for real-time optimization

. Chachuat a,∗, B. Srinivasan b, D. Bonvin c

Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, CanadaDépartement de Génie Chimique, École Polytechnique de Montréal, C.P. 6079 Succ. centre ville, Montréal, QC H3C 3A7, CanadaLaboratoire d’Automatique, École Polytechnique Fédérale de Lausanne (EPFL), Station 9, CH-1015 Lausanne, Switzerland

r t i c l e i n f o

rticle history:eceived 13 October 2008eceived in revised form 23 February 2009ccepted 28 April 2009vailable online 9 May 2009

a b s t r a c t

Challenges in real-time process optimization mainly arise from the inability to build and adapt accuratemodels for complex physico-chemical processes. This paper surveys different ways of using measure-ments to compensate for model uncertainty in the context of process optimization. Three approachescan be distinguished according to the quantities that are adapted: model-parameter adaptation updatesthe parameters of the process model and repeats the optimization, modifier adaptation modifies the

eywords:easurement-based optimization

eal-time optimizationatch-to-batch optimizationlant–model mismatch

constraints and gradients of the optimization problem and repeats the optimization, while direct inputadaptation turns the optimization problem into a feedback control problem and implements optimalityvia tracking of appropriate controlled variables. This paper argues in favor of modifier adaptation, since ituses a model parameterization and an update criterion that are well tailored to meeting the KKT condi-tions of optimality. These considerations are illustrated with the real-time optimization of a semi-batch

odel adaptationodel parameterization

reactor system.

. Introduction

Optimization of process performance has received attentionecently because, in the face of growing competition, it representshe natural choice for reducing production costs, improving prod-ct quality, and meeting safety requirements and environmentalegulations. Process optimization is typically based on a process

odel that is used by a numerical procedure for computing theptimal solution. In practical situations, however, an accuraterocess model can rarely be found with affordable effort. Uncer-ainty results primarily from trying to fit a model of limitedomplexity to a complex process system (Bonvin, 1998; Forbes

Marlin, 1994). The model-fitting task is further complicated byhe fact that process data are usually noisy and signals often doot carry sufficient information for efficient process identificationYip & Marlin, 2004). Therefore, optimization using an inaccurate

odel might result in suboptimal operation or, worse, infeasibleperation when constraints are present.

Two main classes of optimization methods are available for han-ling uncertainty. The main difference relates to whether or not

easurements are used in the calculation of the optimal strategy.

n the absence of measurements, robust optimization is typicallysed, whereby conservatism is introduced to guarantee feasibil-

ty for the entire range of expected variations (Mönnigmann &

∗ Corresponding author. Tel.: +1 905 525 9140x24703; fax: +1 905 521 1350.E-mail address: [email protected] (B. Chachuat).

098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2009.04.014

© 2009 Elsevier Ltd. All rights reserved.

Marquardt, 2003). When measurements are available, adaptive opti-mization, also called real-time optimization (RTO), can help adjust toprocess changes and disturbances, thereby reducing conservatism(Francois, Srinivasan, & Bonvin, 2005). It is interesting to note thatthe above classification is similar to that found in control problemswith the robust and adaptive techniques.

An optimal solution has to be feasible and, of course, opti-mal. In practice, feasibility is typically given a higher prioritythan optimality. In the presence of model uncertainty, feasibil-ity is usually enforced by the introduction of back-offs from theconstraints. The availability of measurements helps reduce theseback-offs and thus improves performance (Chachuat, Marchetti,& Bonvin, 2008). Generally, it is easier to measure or infer con-strained quantities (e.g., temperature or pressure) than estimategradients of the cost and constrained quantities. These elementsclearly set a priority of actions in the framework of adaptive opti-mization.

This paper discusses three adaptive optimization approachesthat differ in the way adaptation is performed, namely (i) model-parameter adaptation, where the measurements are used to refinethe process model, and the updated model is used subsequentlyfor optimization (Chen & Joseph, 1987; Marlin & Hrymak, 1997);(ii) modifier adaptation, where modifier terms are added to the cost

and constraints of the optimization problem, and measurementsare used to update these terms (Forbes & Marlin, 1994; Gao & Engell,2005; Roberts, 1979; Tatjewski, 2002); and (iii) direct input adap-tation, where the inputs are adjusted via feedback control, hencenot requiring on-line numerical optimization but off-line controller

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558 B. Chachuat et al. / Computers and C

esign (Francois et al., 2005; Skogestad, 2000; Srinivasan, Primus,onvin, & Ricker, 2001).

These approaches are reviewed in the first part of the paper.critical discussion follows, which argues in favor of modifier-

daptation methods. It is shown that the modifier-adaptationethods do indeed share the advantages of the other methods, i.e.,

n converging to a KKT point for the plant and also in exhibiting fastonvergence since it is model-based. These issues are illustratedhrough the case study of a semi-batch reactor system.

. Static optimization problem

For continuous processes operating at steady-state, optimiza-ion typically consists in determining the operating point thatptimizes process performance such as minimization of operatingost or maximization of production rate, while satisfying a num-er of constraints (such as bounds on process variables or productpecifications). In mathematical terms, this optimization probleman be stated as follows:

minimize :�

�p(�) := �(�,yp)

subject to : Gp(�) := g(�,yp) ≤ 0(1)

here � ∈Rn� and yp ∈Rny stand for the process input (set point)nd output vectors, respectively; � : Rn� × Rny → R is the planterformance index; and g : Rn� × Rny → R

ng is the vector of con-traints imposed on the input and output variables. It is assumed,hroughout this paper, that � and g are known and the outputariables yp can be either measured or estimated.

In contrast to continuous processes, the optimization of batchnd semi-batch processes consists in determining time-varyingontrol profiles, u(t). This typically involves solving a dynamic opti-ization problem, possibly with path and terminal constraints. A

ractical way of solving such problems is by parameterizing theontrol profiles using a finite number of parameters �, e.g., a poly-omial approximation of u(t) on finite elements. Although therocess is dynamic in nature, a static map can then be used toescribe the relationship between the process inputs � and theutcome of the batch yp. Hence, the problem can be regarded asfinite-dimensional static optimization problem similar to (1),

nd the optimization approaches discussed in the following sec-ions can also be used in the framework of run-to-run optimizationf batch and semi-batch processes (Francois et al., 2005). Thepproach will be illustrated in Section 5.

In practice, the mapping relating the process inputs and outputss typically unknown, and only an approximate model is available,.g., in the form:

= f (�, �), (2)

ith y ∈Rny representing the model outputs, � ∈Rn� the modelarameters, and f : Rn� × Rn� → R

ny the input-output mapping.ccordingly, an approximate solution of Problem (1) is obtainedy solving the following model-based optimization problem:

minimize:�

�(�, �) := �(�, f (�, �))

subject to : G(�, �) := g(�, f (�, �)) ≤ 0.(3)

his formulation assumes that the functions � and g are knownnd independent of the model parameters �; the dependency in �f these functions is implicit via the model outputs y.

If the objective and constraint functions in (1) and (3) are contin-

ous and the feasible domains of these problems are nonempty andounded, (globally) optimal solution points �∗

p and �∗ are guaran-eed to exist for (1) and (3), respectively. Note that there may alsoe multiple local optima to these problems due to nonconvexity.rovided that the active constraints satisfy a regularity condition at

l Engineering 33 (2009) 1557–1567

�∗p and �∗, the KKT conditions – also known as the first-order nec-

essary conditions of optimality (NCO) – must hold at such optimalpoints (Bazaraa, Sherali, & Shetty, 1993). For Problem (3), the KKTconditions read:

Gk(�∗, �) ≤ 0, �∗k

≥ 0, k = 1, . . . , ng,

∂�

∂�(�∗, �) +

ng∑k=1

�∗k

∂Gk

∂�(�∗, �) = 0,

�∗kGk(�∗, �) = 0, k = 1, . . . , ng.

(4)

where �∗ ∈Rng is the vector of KKT multipliers. The KKT conditionsinvolve the quantities G1, . . . , Gng , ∂G1/∂�, . . . , ∂Gng /∂�, ∂�/∂�,which are denoted collectively by the vector C∈RnC subsequently,with nC := ng + n�(ng + 1).

The first step in satisfying the NCO for a given problem consists indetermining the active set. This discrete decision can be taken basedon the sign of the KKT multipliers. Once the active set – representedby Ga – is known, the multipliers can be eliminated from (4), therebyleading to two sets of conditions:

Ga = 0, ∇r� := ∂�

∂�

[I−(

∂Ga

∂�

)+(∂Ga

∂�

)]= 0. (5)

The former guarantees that the active constraints are met,whereas the latter forces the reduced gradient to 0. In real-timeoptimization, these two sets of conditions can be treated using quitedifferent techniques. This important aspect will be discussed in thenext section.

3. Classification of real-time optimization schemes

RTO improves process performance iteratively by adjustingselected optimization variables using measurement data. The goalof this closed-loop adaptation is to drive the operating pointtowards the actual plant optimum in spite of inevitable structuraland parametric model mismatch. RTO methods can be classified indifferent ways. This paper proposes a classification based on thetype and the objective of such adaptation.

The three columns in Fig. 1 differ in the way the adaptationis conducted. In model-parameter adaptation methods, a first-principles model is used, the parameters of which are updatedbased on measurement data; in turn, the updated model is used tocompute the optimal inputs. The middle column corresponds to theupdate of so-called modifiers, which capture the difference betweenthe NCO predicted by the model and those of the plant; that is, afirst-principles model is used for prediction, but its parameters arenot adapted. In the third column, finally, the inputs are adapteddirectly via tracking of an appropriate set of controlled variables.

The rows of the classification in Fig. 1, on the other hand, cor-respond to the two NCO parts discussed in Section 2, namely theactive constraints and the reduced gradient. The former relates tothe problem of feasibility, while the latter relates to the issue of opti-mality. In practical applications, guaranteeing feasible operation istypically of higher importance than achieving the best possible per-formance. Interestingly, the results of a variational analysis in thepresence of small parametric error support the priority given toconstraint satisfaction over the gradient part of the NCO (Chachuatet al., 2008).

3.1. Model-parameter adaptation

The standard way of implementing RTO is via the so-called two-step approach (Chen & Joseph, 1987; Marlin & Hrymak, 1997), alsoreferred to as repeated identification and optimization in the litera-ture. In the first step, the values of (a subset of) the adjustable model

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Fig. 1. Classification of real-time optimization approac

arameters � are estimated using the process output measurements

p. This is typically done by minimizing the lack of closure in theteady-state model Eq. (2). In the second step, the updated model issed to determine a new operating point by solving an optimizationroblem such as (3).

Model-parameter adaptation schemes can be written generi-ally using the following two equations (see Fig. 2):

k = �ad(yp(�k), �k), (6)

k+1 = �opt(�k) (7)

here �ad : Rny × Rn� → Rn� is the map describing model-

arameter adaptation; and �opt : Rn� → Rn� , the map describing

he optimization step. Note that the handles for correction, i.e., thedjustable model parameters �, are a subset of the model parame-ers. A possible adaptation strategy is to minimize the weighted sumf squared errors between the measured and predicted outputs,

k = argmin�

‖yp(�k) − f(�k, �)‖.

In practice, the results of both the identification and optimiza-ion steps clearly need to be diagnosed before implementation; forxample, Miletic and Marlin (1998) have proposed to implementnly significant changes from current operation in the sense of atatistical test.

A key, yet difficult, decision in model-parameter adaptation iso select the parameters to be adapted. These parameters should

e identifiable, represent actual changes in the process, and con-ribute to approach the process optimum; also, model adequacyroves to be a useful criterion to select candidate parameters fordaptation (Forbes, Marlin, & MacGregor, 1994). Clearly, the smallerhe subset of parameters, the better the confidence in the param-

Fig. 2. Optimization via model-parameter adaptation: two-step approach.

sed on adaptation strategy, feasibility and optimality.

eter estimates, and the lower the required excitation. But too fewadjustable parameters can lead to completely erroneous models,and therefore to a false optimum. The question of how plant exper-iments should be designed to contain sufficient information hasbeen addressed, e.g., by Yip and Marlin (2003).

It is well known that the interaction between the model-parameter adaptation and reoptimization steps must be consideredcarefully for the two-step approach to achieve optimality. If themodel is structurally correct and the parameters practically iden-tifiable, convergence to the plant optimum can be achieved inone iteration. However, in the presence of plant/model mis-match, whether the scheme converges, or to which point itconverges, becomes anybody’s guess. This is due to the fact thatthe adaptation objective might be unrelated to the cost andconstraints in the optimization problem; hence, minimizing themean-square error in y may not help in our quest for feasibility andoptimality.

Model flexibility is a critical issue with model-parameter adapta-tion methods. As far as feasibility is concerned, the selected modelparameterization should permit sufficient flexibility for the mea-sured and predicted constraint values to match. Since the constraintexpressions g are independent of �, this latter requirement is auto-matically enforced via output matching, i.e., y = yp. In the caseof ny independent outputs, a necessary condition for feasibilityis therefore that the number of adjustable parameters be greaterthan the number of outputs, n� ≥ ny. In terms of optimality, theselected model parameterization should permit sufficient flexi-bility for component-wise matching of C and Cp. To achieve this,matching of y and yp alone is no longer sufficient, and it mustbe complemented with matching of the output derivatives, i.e.,∂y/∂� = ∂yp/∂�. Provided that all the outputs and output deriva-tives are independent, a model parameterization with at leastny(n� + 1) degrees of freedom is then necessary to achieve optimal-ity. With such a parameterization, it would also become possible tomodify the identification map �ad in (6), e.g., as the minimizationof the weighted sum of squared errors between Cp and C, therebyexplicitly enforcing KKT matching.

Convergence under plant–model mismatch has been addressedby several authors (Biegler, Grossmann, & Westerberg, 1985; Forbeset al., 1994). It has been shown that optimal operation is reached ifmodel adaptation leads to matched KKT conditions for the modeland the plant.

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Convergence result 1 Let the model-parameter adaptation (6)e such that the quantities C predicted by the model match thelant measurements Cp. Then, upon convergence, the model-parameterdaptation scheme (6)–(7)reaches a KKT point for the plant.

.2. Modifier adaptation

In order to overcome plant–model mismatch, several variantsf the two-step approach have been presented in the literature.enerically, they consist in modifying the cost and constraints of theptimization problem in such a way that the KKT conditions of theodel and the plant can match. Letting �k denote the actual oper-

ting point, the optimization problem with modifiers to be solveduring the next RTO execution is given by

minimize:�

�m(�, �) := �(�, �) + ˚T(� − �k)

subject to: Gm(�, �) :=G(�, �) + εG + GT(� − �k) ≤ 0,

(8)

here εG ∈Rng is the constraint-value modifier, G ∈Rn�×ng

he constraint-gradient modifier, and ˚ ∈Rn� the cost-gradientodifier; these modifiers will be denoted collectively by � =

εG1 , . . . , εGng , �G1 T, . . . , �Gng T, �˚T]T

∈RnC subsequently.

The constraint-value modifier represents the difference betweenthe experimental and predicted constraints,

εGi = [Gp,i(�) − Gi(�, �)]�=�k

, i = 1, . . . , ng. (9)

Adapting only εG leads to the so-called constraint-adaptationscheme (Chachuat et al., 2008; Forbes & Marlin, 1994). Such ascheme is rather straightforward and corresponds to commonindustrial practice (Marlin & Hrymak, 1997).The cost-gradient modifier represents the difference between theexperimental and predicted values of the cost gradient,

˚ =[

∂�p

∂�(�) − ∂�

∂�(�, �)

]�=�k

. (10)

The pertinent idea of adding a gradient modifier to the costfunction of the optimization problem dates back to the work ofRoberts (1979). Note that it was originally proposed in the frame-work of two-step methods to better integrate the model updateand optimization subproblems, which has led to the so-calledISOPE approach (see Brdys & Tatjewski, 2005 for an overview). Theidea of modifying the cost gradient without the need to estimatethe model parameters was introduced by Tatjewski (2002).The constraint-gradient modifier, finally, represents the differ-ence between the experimental and predicted values of theconstraint gradients,

�Gi =[

∂Gp,i

∂�(�) − ∂Gi

∂�(�, �)

]�=�k

, i = 1, . . . , ng. (11)

The idea of adding such a first-order modifier term to the process-dependent constraints, in addition to the constraint bias εG, wasproposed recently by Gao and Engell (2005). This modificationpermits the matching, not only of the constraint values, but alsothe constraint gradients.

The update laws in modifier-adaptation schemes can be writtenenerically as (see Fig. 3):

k = �ad(Cp(�k) − C(�k, �)), (12)

k+1 = �opt(

�k

)(13)

here �ad : RnC → RnC stands for the modifier-adaptation map;

nd �opt : RnC → Rn� , the optimization map.

Fig. 3. Optimization via modifier adaptation: matching the KKT conditions.

The handles for correction are now the modifier terms � insteadof the parameters � used in the context of model-parameter adap-tation. The simplest adaptation strategy is to calculate the modifiersby making the full correction (9)–(11) at each RTO execution, i.e.,�k = Cp(�k) − C(�k, �). However, this might lead to excessive cor-rections when operating far away from the optimal operating pointand might make modifier adaptation very sensitive to measure-ment noise. A better strategy consists in filtering the modifiers, e.g.,with a first-order exponential filter (Marchetti, Chachuat, & Bonvin,in press):

�k = (I− K)�k−1 + K[Cp(�k) − C(�k, �)], (14)

where K ∈RnC×nC is a (nonsingular) gain matrix.The experimental quantities Cp required for adaptation are

directly related to the KKT conditions. In practice, the outputmeasurements yp can be used to estimate these quantities. Theexperimental constraint values Gp(�) are readily estimated asg(�, yp). This way, feasibility is enforced, though no guarantee canbe given regarding the correct determination of the active set. Onthe other hand, estimating the experimental cost and constraintgradients, (∂�p/∂�)(�) and (∂Gp/∂�)(�), is more involved and isone of the major bottlenecks in the implementation of modifier-adaptation schemes.

Observe the one-to-one correspondence in the adaptation law(14) between the number of components in (Cp − C) and the num-ber of adjustable parameters in �. Since the Jacobian of (Cp − C)with respect to the modifiers is the gain matrix K (a full-rankmatrix), identifiability is automatically verified, and so are theKKT-matching conditions upon convergence. These nice theoreticalproperties are formalized next.

Convergence result 2 Let the cost and constraint functions beparameterized as in Problem (8). Let also the information on the val-ues of Cp be available and used to adapt the modifiers �. Then, uponconvergence, the modifier-adaptation scheme (12) and (13) reaches aKKT point of the plant.

3.3. Direct input adaptation

This last class of methods provides a way of avoiding repeatednumerical optimization by transforming the optimization prob-lem into a feedback control problem that calculates (set pointsfor) the input variables. This is motivated by the fact that practi-tioners like to use feedback control of selected variables as a wayto counteract plant–model mismatch and plant disturbances, dueto its simplicity and reliability compared to on-line optimization.The challenge is to find functions of the measured variables which,when held constant by adjusting the input variables, enforce opti-mal plant performance (Morari, Stephanopoulos, & Arkun, 1980;Skogestad, 2000). In other words, the goal is to choose appropriatecontrolled variables in order to achieve a similar steady-state per-

formance as would be realized by a (fictitious) on-line optimizingcontroller; these variables are denoted by Zp subsequently, and thecorresponding references by Zref.

In self-optimizing control (Skogestad, 2000), the idea is to use aprocess model to select linear combinations of the output variables,

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ig. 4. Optimization via direct input adaptation: tracking appropriate controlledariables using feedback control.

he tracking of which results in good performance:

p = Myp,

here M is an appropriately chosen matrix. In particular, differ-nt methodologies have been investigated for the choice of M,hich are based on (near) invariance in the presence of uncertainty

Govatsmark & Skogestad, 2005).However, tracking linear combinations of the output variables,

lthough very convenient, hardly provides any guarantee that alant optimum is reached upon convergence. To circumvent thisifficulty, a rather natural idea is to choose the controlled variablesp as the NCO components,

p =[Ga

p ∇r�p

]T, (15)

ith the corresponding set points Zref = 0, thereby enforcing thelant NCO. Two classes of approaches fall within this category:xtremum-seeking control (Ariyur & Krstic, 2003; Guay & Zhang,003), and NCO tracking (Francois et al., 2005; Srinivasan, Biegler,Bonvin, 2008).Perhaps the main difficulty with direct adaptation methods lies

n the fact that the tracked variables do change with the activeet. Also, detection of such a change needs to be dealt with sep-rately from the tracking problem. One possibility is to use awitching strategy, whereby all constraints and KKT multipliersre monitored: a constraint is included in the active set when its binding, and deactivated when the associated KKT multiplierecomes negative (Woodward, Perrier, & Srinivasan, 2007). Anotherpproach consists in precomputing regions that correspond toifferent sets of active constraints based on a (steady-state) pro-ess model (Manum, Narasimhan, & Skogestad, 2008). Since thesewitching approaches render the scheme more complex, it is oftenssumed, in the interest of simplicity, that the active constraints doot change. Note that such an assumption is typically verified formall parametric variations in the case of a strict minimum and isbserved in many practical situations.

Input-adaptation schemes obey the following equations (see

ig. 4):

p(�k) = �act(yp(�k)), or Zp(�k) = �act(Cp(�k)), (16)

k+1 = �con(Zp(�k)). (17)

able 1omparison of various adaptation strategies for real-time optimization.

Model-parameter adapta

djustable parameters �[n�]eal-time information a yp[ny]daptation criterion for adjustable parameters min

�‖yp − y‖

pdate of RTO inputs Optimizationegularization External filterctive set detection Optimizerequirement for feasibility Model flexibility,Ga =Ga

p

equirement for optimality Model flexibility, C = Cp

a Either measured or estimated.b For the case of a controller tracking the NCO components (15).

l Engineering 33 (2009) 1557–1567 1561

Here �act is the map describing the selection of controlled vari-ables, including active set detection (switching logic). Note thatonly the plant outputs yp(�k) are needed to define this map in self-optimizing control, whereas all KKT quantities Cp(�k) are requiredfor tracking the NCO components in (15). �con : Rn� → R

n� is themap describing the square multivariable controller. A simple choicefor this map is in the form of a discrete integral-type controller,

�k+1 = �k + K(Zref − Zp(�k)), (18)

where K ∈Rn�×n� is a gain matrix. Note that, unlike optimization-based schemes, the required regularization is provided naturallyvia appropriate controller tuning.

A crucial question in practice regards the design of the multi-variable controller �con. Since the constrained quantities can oftenbe estimated reliably from the output measurements yp, adjustingthe inputs to track the active constraints is rather straightforward(Bonvin & Srinivasan, 2003; Srinivasan et al., 2001; Stephanopoulos& Arkun, 1980). On the other hand, gradient estimation is moreinvolved and, furthermore, may not lead to significant performanceimprovement. Hence, an interesting option is to devise a two-time-scale adaptation strategy, wherein adaptation of the reducedgradient is implemented at a much slower rate than that of theactive constraints (Francois et al., 2005).

Similar to modifier adaptation, direct input adaptation also pos-sesses nice theoretical properties; for example, when the controllertracks the NCO components (15), the following convergence resultholds:

Convergence result 3 Let the information on the values of Cp beavailable and used to select the active set and track the variables Zp toZref = 0. Then, upon convergence, the input-adaptation scheme (16)and (17)reaches a KKT point of the plant.

4. Comparison of approaches

In this section, we take a critical look at the three aforemen-tioned classes of adaptive optimization schemes. Table 1 comparesthese approaches in terms of various criteria. It is interesting tosee that modifier adaptation can be positioned between model-parameter adaptation and direct input adaptation; several featuresare shared between the first and second columns, while otherfeatures are shared between the second and third columns. Thiscomparison shall lead us to argue in favor of modifier-adaptationmethods, in the sense that they provide a parameterization and anupdate criterion that are well tailored to KKT matching.

The methods differ greatly in the adjustable parameters and thereal-time information (either measured or estimated) that is usedto make the adaptation. Model-parameter adaptation considers the

parameterization � and the output measurements yp, which makesKKT matching difficult to achieve in general. In contrast, modifieradaptation resolves the challenging task of selecting the adjustableparameters by introducing the modifiers � as handles and assum-ing that the KKT-based quantities Cp are available in real-time. Since

tion Modifier adaptation Direct input adaptation

�[nC] —Cp[nC] Cp[nC] b

� = Cp − C —

Optimization ControlExternal filter Controller tuningOptimizer Switching logic– – b

– – b

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specified threshold values:

maxF(t) JA := cC(tf)V(tf) (24)

Table 2Parameters values and initial conditions.

Parameter Value Concentration Value

562 B. Chachuat et al. / Computers and C

he number of components in Cp equals the number of handles, aquare adaptation problem is obtained, and KKT matching is read-ly enforced upon convergence. In direct input adaptation, therere no adjustable parameters as the handles are the input variables

themselves; for the case of NCO tracking, convergence to a KKToint again requires that all Cp components be available.

Major differences also exist in terms of the adaptation crite-ion for the adjustable parameters. Model-parameter adaptationroceeds by solving an optimization problem. Sufficient model flex-

bility (see Section 3.1) is needed to bring y exactly to yp, whichequires at least as many adjustable parameters as there are mea-urements, n� ≥ ny, in the case of independent outputs. Moreover,or the parameter estimates to be accurate, the available measure-

ents must contain sufficiently rich information. Note that oneould consider the alternative criterion min

�‖Cp − C‖ to facilitate KKT

atching. However, with the available n� degrees of freedom, it maye even more difficult to bring the nC components of C exactly to Cp.

n modifier adaptation, one deals with a square adaptation problem,herein the handles � are essentially decoupled and can be read-

ly computed from Cp and C. In direct input adaptation, finally, thedaptation corresponds to a square multivariable control problem.difficulty therein is that a natural pairing between the manipu-

ated variables � and the controlled variables Zp may not exist, thusomplicating the task of designing the RTO controller. The foregoingonsiderations lead us to argue that modifier-adaptation schemesossess the adequate parameterization and use the adequate updateriterion for real-time optimization.

It should also be noted that direct input adaptation differs fromhe other two adaptation schemes in that an optimization problems not solved at each iteration, thus removing much of the on-lineomputational burden and complexity. On the other hand, it cane argued that the use of an on-line optimizer helps determine thective set and alleviates the complicated task of devising a multi-ariable controller at the RTO level. Also, faster convergence may bechieved with an on-line optimizer, although some amount of fil-ering (regularization) is needed to reduce sensitivity to noise anduarantee stability in the presence of plant–model mismatch.

In terms of constraint satisfaction, the ability of model-arameter adaptation to yield a feasible operating point is directly

inked to model flexibility. With modifier adaptation, on the otherand, feasibility is automatically guaranteed upon convergences long as the active process constraints Ga

p can be measuredr estimated. With input adaptation, finally, feasibility is triviallystablished when either the active set remain unchanged withncertainty or active set changes can be detected; otherwise, track-

ng an inappropriate set of constraint may lead to infeasibility.Last but not least, the need for experimental gradient informa-

ion is an important element of comparison. Modifier adaptationnd direct input adaptation typically make use of experimental gra-ients, while model-parameter adaptation does not. In the formerchemes, gradient information is key to achieving feasible and opti-al operation despite model mismatch and process disturbances.

everal techniques for estimating the plant gradients have beenroposed, which differ in terms of their relying on a model orot, as well as their use of steady-state vs. transient measurementata; these techniques, which greatly vary in accuracy too, include:a) finite differences (Roberts, 1979), (b) black-box identificationBamberger & Isermann, 1978, Golden & Ydstie, 1989; Mansour &llis, 2003; Zhang & Forbes, 2006), (c) dither-signal superpositionAriyur & Krstic, 2003; Guay & Zhang, 2003, (d) linearization (neigh-

oring extremals) (Gros, Srinivasan, & Bonvin, 2009), (e) Broydenethods (Gao & Engell, 2005; Mansour & Ellis, 2003), and (f) use

f multiple parallel units (Srinivasan, 2007; Woodward, Perrier, &rinivasan, 2009). Which approach to use is clearly problem depen-ent and lies beyond the scope of this paper.

l Engineering 33 (2009) 1557–1567

5. Case study

The considerations discussed in the previous sections of thepaper are now illustrated for the run-to-run optimization of a semi-batch reactor process. This problem is recast as a static optimizationproblem via parameterization of the time-varying control profile,as discussed in Section 2.

The reaction system is the acetoacetylation of pyrrole withdiketene and consists of 4 reactions:

A + Bk1−→C

2Bk2−→D

Bk3−→E

C + Bk4−→F,

with A: pyrrole; B: diketene; C: 2-acetoacetyl pyrrole; D:dehydroacetic acid; E: oligomers; F: undesired by-product. A first-principles model for the semi-batch reactor is as follows:

cA = −k1cAcB − F

VcA (19)

cB = −k1cAcB − 2k2c2B − k3cB − k4cBcC + F

V(cin

B − cB) (20)

cC = k1cAcB − k4cBcC − F

VcC (21)

cD = k2c2B − F

VcD (22)

V = F, (23)

where cA, cB, cC and cD stand for the concentrations of species A, B,C and D, respectively; V, the reactor volume; F , the inlet flow rateof species B; and cin

B , the concentration of B in the feed. The initialvalues for the state variables are reported in Table 2. See Ruppen,Bonvin, and Rippin (1998) for a detailed description of the processand model.

Throughout this case study, the full reaction mechanism is con-sidered for the simulated plant, with the corresponding kineticparameter values given in Table 2. On the other hand, only the firsttwo reactions (A + B → C and 2B → D) are taken into account inthe plant model, i.e., the remaining two reaction are considered tobe unknown side reactions (k3 = k4 = 0). This way, the model hasstructural mismatch.

5.1. Optimization Problem A

The first problem consists in determining the feed profile ofspecies B that maximizes the number of moles of C at final time,while keeping the concentrations of B and D at terminal time below

k1 0.053 L mol−1 min−1 cA(0) 0.72 mol L−1

k2 0.128 L mol−1 min−1 cB(0) 0.05 mol L−1

k3 0.028 min−1 cC(0) 0.08 mol L−1

k4 0.001 L mol−1 min−1 cD(0) 0.01 mol L−1

cinB 5 mol L−1 V(0) 1 L

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B. Chachuat et al. / Computers and Chemical Engineering 33 (2009) 1557–1567 1563

F s. Dashv

w2D

tTsafwstsastota

mttJ

tssstaGst

t�u

p(see Table 1).

Modifier adaptation The optimization problem (24) is modi-fied by introducing the constraint-value modifiers εB and εD, whilethe cost and constraint gradients remain uncorrected. In the k th

ig. 5. Optimal operation in Problem A for two different control parameterizationariable-length control stages.

s.t. Model (19 − 23)

cB(tf) ≤ cmaxB

cD(tf) ≤ cmaxD

0 ≤ F(t) ≤ Fmax,

ith the final time tf = 250 min; the maximal inlet flow rate Fmax =× 10−3 L min−1; and the maximal concentrations of species B andat final time cmax

B = 0.025 mol L−1 and cmaxD = 0.15mol L−1.

The optimal control and state trajectories, as obtained fromwo different control parameterizations, are shown in Fig. 5.he piecewise-constant parameterization over 100 constant-lengthtages shows that the optimal feed profile consists of 3 parts orrcs: (i) a first arc where F = Fmax; (ii) a second arc where theeed level is intermediate between 0 and Fmax; and (iii) a last arc

here the feed rate is equal to 0. Moreover, both terminal con-traints are active in the optimal solution. Also shown in Fig. 5 arehe results for a control parameterization with 3 variable-lengthtages. With this parameterization, the input (decision) variablesre � := (tm Fs ts)T with tm, the switching time between the first andecond arcs; Fs, the (constant) feed rate along the second arc; and ts,he switching time between the second and third arcs. The valuesf the objective function for both parameterizations are very close,hus indicating that this latter parameterization yields an adequatepproximation.

Solving the discretized optimal control problem using the plantodel (i.e., with k3 = k4 = 0) gives the same optimal solution struc-

ure (3 arcs, 2 active terminal constraints). Open-loop application ofhis solution, although feasible, yields a much worse performanceAol ≈ 0.3874 mol compared to JA

opt ≈ 0.5079 mol.The use of various real-time optimization techniques to reduce

his large optimality gap is investigated next. Both terminal con-traints being active in the optimal solution, only constraintatisfaction schemes are considered, i.e., without cost and con-traint gradient correction. It is assumed that measurements forhe reactor volume and the concentrations of species B, C and Dre available at the end of the batch. Moreover, a 5% zero-meanaussian multiplicative noise is added to the concentration mea-

urements, in response to which a 10% back-off is defined for the

erminal constraints in the optimization problem.

Model-parameter adaptation The adjustable model parame-ers correspond to the kinetic coefficients of the first two reactions,:= (k1, k2). At the end of the k th RTO execution, their values arepdated by minimizing a weighted sum of squared errors between

-dotted line: CVP with 100 constant-length control stages. Solid line: CVP with 3

the measured and predicted concentrations of species B, C and D,

(k∗1, k∗

2) = argmin�

{ ∑i ∈ {B,C,D}

(1 − ci(tf; �)

cp,i(tf)

)2

�=�k

}, (25)

with cp,i(tf) denoting the concentration of species i measured at theend of the batch. To avoid parameter corrections that would be tooaggressive and reduce sensitivity to measurement noise, the fittedkinetic coefficients are filtered via a simple exponential filter,

(k1,k

k2,k

)= (1 − ��)

(k1,k−1k2,k−1

)+ ��

(k∗

1k∗

2

)

with the filter parameter �� ∈ (0, 1]. In turn, the filtered kineticcoefficients are used to update the control variables �k+1 =(tm,k+1 Fs,k+1 ts,k+1)T by solving the optimization problem (24).

The results for this adaptation scheme, with �� = 1.0 and 0.5,are shown in the upper row of Fig. 6. Observe first that the ter-minal constraint on cB is slightly violated when no filtering isused (�� = 1), but this violation can be avoided through filter-ing (�� = 0.5). In terms of performance, some improvement isobtained – the converged cost value being around JA

par ≈ 0.43 mol–, yet an important optimality gap persists. In particular, theleftmost plot indicates that the adapted control profile remainsignificantly different from the optimal profile, especially the con-trol value Fs. The difficulty in this case is linked to the fact thatthe model parameters � do not permit sufficient flexibility forthe model to match the experimental measurements; in particu-lar, we have n� = 2 < ny = 3 here. A way to circumvent this lackof flexibility would be, e.g., to add a fictitious reaction to theplant model and estimate the additional kinetic coefficient so thatn� = ny = 3. Another way to guarantee feasibility upon conver-gence would be to not take the measurement of cC into accountin the estimation problem (25) so as to force the condition Ga = Ga

RTO execution, the updated control variables �k are determined bysolving the optimization problem:

maxtm,Fs,ts

cC(tf)V(tf) (26)

Page 8

1564 B. Chachuat et al. / Computers and Chemical Engineering 33 (2009) 1557–1567

F meter

Ne(

w

mFitia

ig. 6. Application of real-time optimization to Problem A. Upper plots: model-para

s.t. Model (19 − 23)

cB(tf) + εB,k−1 ≤ cmaxB

cD(tf) + εD,k−1 ≤ cmaxD

0 ≤ Fs ≤ Fmax.

ext, constraint-modifier adaptation proceeds through a first-orderxponential filter,

εB,k

εD,k

)= (1 − �ε)

(εB,k−1

εD,k−1

)+ �ε

(cp,B(tf) − cB(tf)

cp,D(tf) − cD(tf)

)�=�k

,

ith the filter parameter �ε ∈ (0, 1].The results for modifier adaptation, with the initial constraint-

odifier values εB,0 = εB,0 = 0, are shown in the middle row of

ig. 6 for the filter parameters �ε = 1.0 and 0.5. Most of the optimal-ty gap is recovered upon application of this simple scheme, withhe cost value on the order of JA

mod ≈ 0.49 mol. The remaining losss incurred by (i) the back-offs imposed on the terminal constraints,nd (ii) the fact that the cost and constraint gradients have not been

adaptation. Middle plots: modifier adaptation. Lower plots: direct input adaptation.

adapted and thus do not match those of the plant at the convergedoperating point.

It should be noted that implementation of the full modifierscheme has been found to be beneficial only in the case wherereliable gradient estimates are available. In particular, it could beshown that adapting the gradient part is detrimental to the opti-mization objective with a 10% multiplicative white noise in thegradient estimates.

Direct input adaptation Both terminal constraints are bindingaccording to the nominal model-based solution of Problem (24).Two degrees of freedom are thus necessary to track the activeconstraints. In terms of performance, it is found that the occur-rence of the switching time tm has a very limited influence; forinstance, forcing tm = 0 in the parameterization (thereby leavingonly 2 arcs), would result in a performance loss lower than 0.1%.

For simplicity, this switching time is fixed at its nominal value,tm = 4.71 min, subsequently. There results a 2 × 2 control prob-lem, with the manipulated variables (Fs, ts), the controlled variables(cp,B(tf) − cmax

B , cp,D(tf) − cmaxD ), and the set-points (0,0). It is found,

using the relative gain array (RGA) technique (Ogunnaike & Ray,

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B. Chachuat et al. / Computers and Chemical Engineering 33 (2009) 1557–1567 1565

F . Dashv

1ii(

w[

rnmtg

5

o

m

wpT

tTsaasropC

wt

ig. 7. Optimal operation in Problem B for two different control parameterizationsariable-length control stages.

994), that two independent control loops can be devised by pair-ng cp,B(tf) − cmax

B with ts, and cp,D(tf) − cmaxD with Fs. Simple discrete

ntegral controllers are then used for each loop,

ts,k

Fs,k

)=(

ts,k−1Fs,k−1

)−(

�ts

�Fs

)(cp,B(tf) − cmax

Bcp,D(tf) − cmax

D

)�=�k−1

,

ith the controller parameters �ts [L min mol−1] and �Fs

L2 min−1 mol−1].The results of direct input adaptation are given in the lower

ow of Fig. 6, for two different controller settings, starting from theominal values of the control variables. Observe that the perfor-ance and adaptation speed of direct input adaptation are similar

o those of modifier adaptation. Moreover, on-line complexity isreatly reduced in this approach compared to the other schemes.

.2. Optimization Problem B

The second problem differs from Problem A in that large feedsf B are now penalized in the objective function:

axF(t) JB := cC(tf)V(tf) − ω

∫ tf

0

F(t)2dt (27)

s.t. Model (19 − 23)

cB(tf) ≤ cmaxB

cD(tf) ≤ cmaxD

0 ≤ F(t) ≤ Fmax,

ith the penalty coefficient taken as ω = 2500 L2 min−1 mol−1. Thearameters and initial conditions are the same as those given inable 2, and so are the constraint bounds.

The optimal control and state trajectories, as obtained fromwo different control parameterizations, are shown in Fig. 7.he piecewise-constant parameterization over 100 constant-lengthtages shows that the optimal feed profile now consists of one singlerc, and both terminal constraints are inactive. Also shown in Fig. 7re the results of a control parameterization with 2 variable-lengthtages, for which the inputs � := (Fs ts)T consist of a constant flowate Fs followed by a zero flow rate, the switching between the twoccurring at time ts. The values of the objective function for both

arameterizations are very close, thus indicating that this 2-stageVP yields an accurate approximation.

Solving this optimal control problem using the plant model (i.e.,ith k3 = k4 = 0) results in the same structure for the optimal solu-

ion (2 arcs, all terminal constraints inactive); the performance of

-dotted line: CVP with 100 constant-length control stages. Solid line: CVP with 2

the open-loop solution is JBol ≈ 0.2135 mol, i.e., not too far from the

plant optimum JBopt ≈ 0.2179 mol.

Various real-time optimization techniques are investigated nextto reduce this small optimality gap. Because the solution is uncon-strained, the emphasis is on gradient-correction schemes. Thesame measurements as in Problem A are considered. In addi-tion, estimates of the cost derivatives with respect to the controlparameters Fs and ts are calculated numerically for each batch andadding a 10% zero-mean Gaussian multiplicative noise to thesevalues.

Model-parameter adaptation The values of the adjustablemodel parameters k1 and k2 are updated in a similar way as pre-viously in Problem A. The results for this adaptation scheme, with�� = 1.0 and 0.5, are shown in the upper row of Fig. 8. Quite unex-pectedly, model-parameter adaptation in this case results in a lossof performance (JB

par ≈ 0.198 mol) compared to the open-loop nom-inal solution JB

ol. The reason for this behavior is twofold. Firstly,the selected parameter identification criterion does not adequatelymatch the optimization objective; and secondly, the kinetic coeffi-cients k1 and k2 do not allow sufficient flexibility for the model tomatch the experimental data; matching the components of C andCp would require ny(n� + 1) = 12 degrees of freedom, whereas onlyn� = 2 are available.

Modifier adaptation The optimization problem (27) is modi-fied by adding a gradient-modifier term to the cost function; onthe other hand, the constraints are left uncorrected for simplicity,since the optimal solution is known to be unconstrained. The con-trol variables �k applied during the k th RTO execution are obtainedby solving the optimization problem:

maxFs,ts

cC(tf)V(tf) − ωtsF2s + �T

k−1(� − �k−1) (28)

s.t. Model (19 − 20)

cB(tf) ≤ cmaxB

cD(tf) ≤ cmaxD

0 ≤ Fs ≤ Fmax,

where � ∈R2 denotes the vector of cost-gradient modifiers. Adap-tation of these modifiers proceeds as follows:[ ]

�k = (1 − �)�k−1 + �

∂�p

∂�− ∂�

∂��=�k

,

with the filter parameter � ∈ (0, 1] and � := cC(tf)V(tf) − ωtsF2s

denoting the cost function.

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1566 B. Chachuat et al. / Computers and Chemical Engineering 33 (2009) 1557–1567

Fig. 8. Application of real-time optimization to Problem B. Upper plots: model-parameter adaptation. Middle plots: modifier adaptation. Lower plots: direct input adaptation.T

ufittnwp

btl(faob

he actual plant optimum is represented by the symbol ‘©’ on the right plots.

The results for modifier adaptation, with the initial modifier val-es �0 = (0 0)T, are shown in the middle row of Fig. 8 for thelter parameters � = 0.5 and 0.2. This scheme converges quickly

o the actual plant optimum. Since no terminal constraint is active,he residual optimality gap results from the effect of measurementoise only. The experimental gradient estimates being corruptedith large amounts of noise, the use of a small value of the filter

arameter � produces better results in this case.Direct input adaptation The optimal solution of Problem (27)

eing unconstrained, the available degrees of freedom can be usedo force the cost gradient to zero. The resulting 2 × 2 control prob-em has the manipulated variables (Fs, ts), the controlled variables

∂�p/∂Fs, ∂�p/∂ts) and the set-points (0,0). The control map isound to be scale-independent and diagonally dominant via RGAnalysis, which leads to the natural pairing of ∂�p/∂Fs with Fs andf ∂�p/∂ts with ts; yet, some amount of interaction is also detectedetween the two control loops. Simple discrete integral controllers

are used,

(ts,k

Fs,k

)=(

ts,k−1Fs,k−1

)−(

�ts

�Fs

)(∂�p/∂ts

∂�p/∂Fs

)�=�k−1

,

with the controller parameters �ts [min2 mol−1] and �Fs

[L2 min−2 mol−1].The results of direct input adaptation are given in the lower row

of Fig. 8, for two different controller settings. To limit the amount of

interaction between the control loops, the controller Fs → ∂�p/∂Fs

is tuned to proceed at a faster rate than the controller ts → ∂�p/∂ts.The convergence of this direct input-adaptation scheme appears tobe rather slow, even though the iterates eventually reach a neigh-borhood of the actual plant optimum.

Page 11

emica

6

spalulbb

capfinSwmt

A

sM

R

A

B

B

B

B

B

B

C

C

F

F

F

B. Chachuat et al. / Computers and Ch

. Conclusions

This paper provides a classification of real-time optimizationchemes and analyzes their ability to enforce feasible and optimalrocess operation by using on-line measurements. The similaritiesnd differences between the various schemes have been high-ighted, and it has been argued that modifier-adaptation schemesse a model parameterization and an update criterion that are tai-

ored to the matching of KKT conditions. These considerations haveeen illustrated through the run-to-run optimization of a semi-atch reactor system in the presence of structural model mismatch.

The combination of specific features of the various approachesertainly holds much promise towards improved performance ofdaptive optimization. For example, the combination of model-arameter adaptation (which ensures fast convergence for the first

ew iterations and can detect changes in the active set) with directnput adaptation (which provides the necessary gradients in theeighborhood of the plant optimum) has been demonstrated byrinivasan and Bonvin (2003). Another interesting combinationould be to use modifier adaptation at one time scale and performodel-parameter adaptation at a slower rate, thus giving rise to a

wo-time-scale adaptation strategy.

cknowledgments

The authors would like to acknowledge useful discussions oneveral topics of this paper with Alejandro Marchetti of EPFL andoncef Chioua of École Polytechnique, Montréal.

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