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Dynamic Optimization in the Batch Chemical Industry

D. Bonvin, B. Srinivasan, D. Ruppen*

Institut d’Automatique, Ecole Polytechnique Federale de LausanneCH-1015 Lausanne, Switzerland

*Computer-aided Process Engineering, Lonzagroup, CH-3930 Visp, Switzerland

April 6, 2001

AbstractDynamic optimization of batch processes has attracted more attention in recent years since, inthe face of growing competition, it is a natural choice for reducing production costs, improvingproduct quality, and meeting safety requirements and environmental regulations. Since themodels currently available in industry are poor and carry a large amount of uncertainty, standardmodel-based optimization techniques are by and large ineffective, and optimization methods needto rely more on measurements.

In this paper, various measurement-based optimization strategies reported in the literatureare classified. A new framework is also presented, where important characteristics of the optimalsolution that are invariant under uncertainty are identified and serve as references to a feedbackcontrol scheme. Thus, optimality is achieved by tracking with no numerical optimization requiredon-line. When only batch-end measurements are available, the proposed method leads naturallyto an efficient batch-to-batch optimization scheme. The approach is illustrated via simulationof a semi-batch reactor in the presence of uncertainty.

KeywordsDynamic optimization, Optimal control, Batch chemical industry, On-line optimization, Batch-to-batch optimization, Run-to-run optimization.

Introduction

Batch and semi-batch processes are of considerableimportance in the chemical industry. A wide vari-ety of specialty chemicals, pharmaceutical products,and certain types of polymers are manufactured inbatch operations. Batch processes are typically usedwhen the production volumes are low, when isola-tion is required for reasons of sterility or safety, andwhen frequent changeovers are necessary. With therecent trend in building small flexible plants that areclose to the markets of consumption, there has beena renewed interest in batch processing (Macchietto,1998).

From a process systems point of view, the key featurethat differentiates continuous processes from batchand semi-batch processes is that the former have asteady state, whereas the latter are inherently time-varying in nature (Bonvin, 1998). This paper con-siders batch and semi-batch processes in the same

manner and, thus herein, the term ‘batch processes’includes semi-batch processes as well.

The operation of batch processes typically involvesfollowing recipes that have been developed in thelaboratory. However, owing to differences in bothequipment and scale, industrial production almostinvariably necessitates modifications of these recipesin order to ensure productivity, safety, quality, andsatisfaction of operational constraints (Wiederkehr,1988). The ‘educated trials’ method that is oftenused for recipe adjustment is based on heuristics andresults in conservative profiles. Conservatism is nec-essary here to guarantee feasibility despite processdisturbances.

To shorten the time to market (by bypassing an elab-orate scale-up process) and to reduce operationalcosts (by reducing the conservatism), an optimiza-tion approach is called for, especially one that canhandle uncertainty explicitly. Operational decisionssuch as temperature or feed rate profiles are then de-

Dynamic Optimization in the Batch Chemical Industry, Bonvin et. al 2

termined from an optimization problem, where theobjective is of economic nature and the various tech-nical and operational constraints are considered ex-plicitly. Furthermore, due to the repetitive natureof batch processes, these problems can also be ad-dressed on a batch-to-batch basis.

The objectives of this paper are threefold: i) ad-dress the industrial practice prevailing in the batchspecialty chemical industry and discuss the result-ing optimization challenges, ii) review the dynamicoptimization strategies available for batch processes,with an emphasis on measurement-based techniques,and iii) present a novel scheme that uses the availableprocess measurements directly (i.e., without the of-ten difficult step of model refinement) towards thegoal of optimization. Accordingly, the paper hasthree major parts:

• Industrial perspectives in batch processing: Themajor operational challenges aim at speedingup process/product development, increasing theproductivity, and satisfying safety and prod-uct quality requirements (Allgor et al., 1996).These tasks need to be performed in an environ-ment characterized by a considerable amount ofuncertainty and the presence of numerous op-erational and safety-related constraints. Themeasurements available could be used to helpmeet these challenges.

• Optimization strategies for batch processes:These are reviewed and classified according to:i) whether uncertainty is considered explicitly,ii) whether measurements are used, iii) whethera model is used to guide the optimization. Thetype of measurements used for optimization (on-line, off-line) adds another dimension to theclassification.

• Invariant-based optimization scheme: New in-sights into the optimal solution for a class ofbatch processes have led to an alternative wayof dealing with uncertainty. It involves: i)the off-line characterization of the optimal so-lution using a simplified model, ii) the selec-tion of signals that are invariants to uncer-tainty, and iii) a model-free implementationby tracking these invariants using a limitednumber of measurements. This results in amodel-free though measurement-based imple-mentation that is quite robust towards uncer-tainty. An interesting feature of this frameworkis that it permits naturally to combine off-linedata from previous batches with on-line datafrom the current batch.

The paper is organized as follows. The industrialperspectives in batch processing are presented first.The next section briefly reviews the optimizationstrategies available for batch processes and proposesa classification of the methods. The invariant-basedoptimization framework is then developed and an ex-ample is provided to illustrate the theoretical devel-opments. Conclusions are drawn in the final section.

Industrial Perspectives in Batch Pro-cessing

It is difficult to address in generic terms the perspec-tives prevailing in the batch chemical industry sincethe processing environments and constraints differconsiderably over the various activities (specialtychemicals, pharmaceuticals, agro and bio products,etc.). Thus, the situation specific to the produc-tion of intermediates in the specialty chemical in-dustry will be emphasized in this section. The cus-tomer – typically an end-product manufacturer – of-ten generates competition between several suppliersfor the production of a certain product. The suppli-ers need to investigate the synthesis route and designan appropriate production process. The competitionforces the suppliers to come up, under considerabletime pressure, with an attractive offer (price/kg) ifthey want to obtain the major share of the deal.

Batch processes are usually carried out using rela-tively standardized pieces of equipment whose oper-ating conditions can be adjusted to accommodate avariety of products. The working environment thatwill be considered is that of multi-product plants.In a multi-product plant, a number of products aremanufactured over a period of time, but at any giventime, only one product is in the process. The se-quence of tasks to be carried out on each piece ofequipment such as heating, cooling, reaction, dis-tillation, crystallization, drying, etc. is pre-defined,and the equipment item in which each task is per-formed is also specified (Mauderli and Rippin, 1979).

Operational Objectives

The fundamental objective is of economic nature.The investment (in time, personnel, capital, etc.,)should pay off, as the invested capital has to com-pare favorably with other possible investments. Thisfundamental objective can in turn be expressed interms of technical objectives and constraints, whichare presented next.


• Productivity: This is the key word nowadays.However, high productivity requires stable pro-duction so as to reduce the amount of correctivemanual operations that are costly in terms ofproduction time and personnel. Reducing thetime necessary for a given production is partic-ularly interesting when the number of batchesper shift can be increased. In multi-productplants, however, equipment constraints (bottle-necks) and logistic issues often limit productiv-ity.• Product quality: Quality is often impaired by

the appearance of small amounts of undesiredby-products. The presence of impurities (alsodue to recycled solvents) is very critical sinceit can turn an acceptable product into waste.Removing impurities is often not possible orcan significantly reduce throughput. Also, froman operational, logistic and regulation point ofview, it is often not possible to use blending op-erations in order to achieve the desired averagequality.Reproducibility of final product compositiondespite disturbances and batch-to-batch varia-tions is important when the process has to workclosely to some quality limit (for example, whenthe quality limits are tight). Improving the se-lectivity of an already efficient process is oftennot seen as a critical factor. However, whenthe separation of an undesirable by-product isdifficult, the selectivity objective may be quiteimportant.• Safety aspects: The safety aspects (runaway,

contamination, etc.) are of course very impor-tant. Safety requirements can lead to highlyconservative operation. Here, the real obstacleis the lack of on-line information. If informationabout the state of the process were available, theprocess engineer would know how to guaranteesafety or react in the case of a latent problem.Thus, the difficulty results from a measurementlimitation and not from a lack of operationalknowledge.• Time-to-market: The economic performance is

strongly tied to the speed at which a new prod-uct/process can be developed. The product life-time of specialty chemicals is typically shorterthan for bulk chemicals. Since the productionin campaigns reduces the time to learn, it isnecessary to learn quickly and improve the pro-ductivity right away. After a couple of years, aprofitable new product may become a commod-ity (of much lesser value), for which the devel-opment of a second-generation process is often

considered.Nowadays, there is a trend in the specialtychemical industry to skip pilot plant investi-gations unless the process is difficult to scaleup. The situation is somewhat different in phar-maceuticals production, where pilot plant in-vestigations are systematically used since theyalso serve to produce the small ‘first amounts’needed.

Industrial Practice

Though the problem of meeting the aforementionedobjectives could be solved effectively as an optimiza-tion problem, there have been only a few attemptsin industry to optimize operations through mathe-matical modeling and optimization techniques. Therecipes are developed in the laboratory in such a waythat they can be implemented safely in production.The operators then use heuristics gained from expe-rience to adjust the process periodically (wheneverthis is allowed), which leads to slight improvementsfrom batch to batch (Verwater-Lukszo, 1998). Thestumbling blocks for the use of mathematical mod-eling and optimization techniques in industrial prac-tice have been partly organizational and partly tech-nical.

Organizational issues. At the organizationallevel, the issues are as follows:

• Registration: Producers of active compoundsin the food and pharmaceuticals areas have topass through the process of registration with theFood and Drug Administration. Since this is acostly and time-consuming task, it is performedsimultaneously with R&D for a new productionprocess. Thus, the main operational parame-ters are fixed within specified limits at an earlystage of the development. Since the specifica-tions provided by the international standards ofoperation (GMP) are quite tight, there is verylittle room for maneuver left. It is importantto stress that the registration is tied to bothproduct and process.

• Multi-step process: In the R&D phase of a largemulti-step process, different teams work on dif-ferent processing steps. Often, each team triesto optimize its process subpart, thereby intro-ducing a certain level of conservatism to ac-count for uncertainty. Consequently, the result-ing process is the sum of conservatively designedsubparts, which often does not correspond tothe optimum of the global process!


• Role of control and mathematical optimization:In many projects, control is still considered tobe a standard task that has to be performedduring the detailed engineering phase and notas a part of the design phase of the process. It islike ‘painting’ a controller or an optimizer oncethe process has been built. At this late stage,there is so much conservatism and robustnessin the system that it does not require a sophis-ticated control strategy. However, the perfor-mance may still be far from being optimal.

All these organizational problems can be resolvedby resorting to ‘global thinking’. It has become achallenge for both project leaders and plant man-agers to make chemists and engineers think and actin a global way. It is done through fostering in-terdisciplinary teamwork and simultaneous ratherthan sequential work for process research, develop-ment and production (R&D&P). The objective isa globally optimal process and not simply the jux-taposition of robust process subparts. Team workamounts to having R&D&P solutions worked out si-multaneously by interdisciplinary teams consistingof a project leader, chemists, process engineers, pro-duction personnel and specialists for analytics, sim-ulation, statistics, etc.

Technical issues. The main technical issues re-late to modeling and measurements, the presence ofboth uncertainty and constraints, and the proper useof the available degrees of freedom for process im-provement. These are addressed next.

• Modeling: In the specialty chemical industry,molecules are typically more complex than inthe commodity industry, which often results incomplex reaction pathways. Thus, it is illusoryto expect constructing detailed kinetic models.The development of such models may exceedone man-year, which is incompatible with theobjectives of batch processing. So, what is oftensought in batch processing, is simply the abilityto predict the batch outcome from knowledge ofits initial phase.

Modern software tools such as Aspen Plus,PRO/II, or gPROMs have found wide ap-plication to model continuous chemical pro-cesses (Marquardt, 1996; Pantelides and Britt,1994). The situation is somewhat differentin batch specialty chemistry. Though batch-specific packages such as Batch Plus, BATCH-FRAC, CHEMCAD, BatchCAD, or BaSYS areavailable, they are not generally applicable. Es-pecially the two important unit operations, re-

action and crystallization, represent a consider-able challenge to model at the industrial level.

For batch processes, modeling is often done em-pirically using input/output static models onthe basis of statistical experimental designs.These include operational variables specified atthe beginning of the batch and quality vari-ables measured at the end of the batch. Time-dependent variables are not considered beyondvisual comparison of measured profiles. Some-times the model is a set of simple linguisticrules based on experience, e.g. when ‘low’ then‘bad’. Occasionally, the model consists of a sim-ple energy balance, or the main dynamics areexpressed via a few ordinary differential equa-tions. The modeling objective is not accuracybut rather the ability to semi-quantitatively de-scribe the major tradeoffs present in the pro-cess such as the common one between qualityand productivity in many transformation andseparation processes. For example, an increasein reflux ratio improves distillate purity but re-duces distillate flow rate; or a temperature in-crease can improve the yield at the expense ofselectivity in a chemical reaction system.

• Measurements: Quality measurements are typi-cally available at the end of the batch via, for ex-ample, off-line chromatographic methods (GC,HPLC, DC, IC). In addition, physical measure-ments such as temperature, flow, pressure, orpH may be available on-line during the courseof the batch. However, they are rather unspe-cific with respect to the key variables (concen-trations) of the chemical process. Other on-line measurements such as conductivity, viscos-ity, refractive index, torque, spectroscopy, andcalorimetry are readily available in the labora-tory, but rarely in production. Pseudo on-lineGC and HPLC are less effective in batch pro-cessing than with continuous processes due torelatively longer measurement delays.

On-line spectroscopy (FTIR, NIR, Raman) hasopened up new possibilities for monitoringchemical processes (McLennan and Kowalski,1995; Nichols, 1988). These techniques rely onmultivariate calibration for accurate results, i.e.,the spectral measurements need to be calibratedwith respect to known samples containing allthe absorbing species. Though on-line spec-troscopy is getting more common in the labora-tory, the transfer of many measurement systemsfrom the laboratory to the plant is still a realchallenge. For example, many processing steps


deal with suspensions that lead to plugging anddeposition problems. Even if these problemscan be handled at the laboratory scale, theystill represent formidable challenges at the pro-duction level. There is presently a strong pushto develop and validate measurement techniquesthat can work equally well throughout the threelevels of Research, Development and Produc-tion.When quality measurements are not directlyavailable, state estimation (or soft sensing) istypically utilized. However, physical on-linemeasurements are often too unspecific for on-line state estimation in batch processes (e.g.heat balance models are too insensitive withrespect to the chemical transformations of in-terest). Current practice indicates that thereare very few applications of state estimationin specialty chemistry. However, state estima-tion works well in fermentation processes due tothe availability of additional physical measure-ments and the possibility to reconstruct con-centrations without the use of kinetic models(Bastin and Dochain, 1990).

• Uncertainty: Uncertainty is widely present inthe operation of batch processes. Firstly, it en-ters in the reactant quality (changes in feed-stock), which is the main source of batch-to-batch variations. Secondly, uncertainty comesin the form of modeling errors (errors in modelstructure and parameters). These modeling er-rors can be fairly large since, according to thephilosophy of batch processing, little time isavailable for the modeling task. Thirdly, processdisturbances and measurement noise contributeto the uncertainty in process evolution (e.g. un-detected failure of dosing systems; change in the‘quality’ of utilities such as brine temperature,or of manual operations such as solid charge).Recipe modifications from one batch to the nextto tackle uncertainty are common in the spe-cialty chemical industry, but less so for the ex-clusive syntheses in agro and pharmaceuticalsproduction. Uncertainty is typically handledthrough:

– The choice of conservative operation suchas extended reaction time, lower feed rateor temperature, the use of a slightly over-stoichiometric mixture in order to forcethe reaction to fully consume one reactant(Robust mode).

– Feed stock analyses leading to appropri-ate adjustments of the recipe (Feedforward

mode). Adjustment is usually done byscaling linearly certain variables such asthe final time or the dilution, more rarelythe feed rate or the temperature.

– Rigorous quality checks through off-lineanalyses, or the use of standard measure-ments such as the temperature differencebetween jacket and reactor, leading to ap-propriate correction of the recipe (Feed-back mode). For example, a terminal con-straint can be met by successive additionof small quantities of feed towards the endof a batch to bring the reaction to the de-sired degree of completion (Meadows andRawlings, 1991).

The problem of scale-up can also be viewed asone of (model) uncertainty. The data availablefrom laboratory studies do not quite extrapolateto the production level. Thus, when the strate-gies developed in the laboratory are used at theproduction level, they do carry a fair amountof uncertainty. Furthermore, the pressure toreduce costs and to speed up process develop-ment calls for large scale-ups with a consider-able amount of extrapolation. As a result, theproposed strategies can be rather conservative.

• Constraints: Industrial processing is naturallycharacterized by soft and hard constraints re-lated to equipment and operational limitationsand to safety aspects. In batch processing,there is the additional effect of terminal con-straints (selectivity in reaction systems, purityin separation systems, admissible levels of im-purities, etc.). Furthermore, in multi-productbatch production, the process has to fit in anexisting plant. Thus, ensuring feasible opera-tion comes before the issue of optimality, andprocess designers normally introduce sufficientconservatism in their design so as to guaranteefeasibility even in the worst of conditions.

The need to improve performance calls for a re-duction of the conservatism that is introducedto handle uncertainty. Performance improve-ment can be obtained by operating closer toconstraints, which can be achieved by measur-ing/estimating the process state with respect tothese constraints. Riding along an operationalconstraint is often done when the constrainedvariable is directly implemented (such as maxi-mum feed rate) or can be measured (such as atemperature).


• Time-varying decisions: Traditionally, chemistsin the laboratories and operators in the plantswere used to thinking in terms of constantvalues (experimental planning results in staticmaps between design variables and process per-formance). New sensors and increasing com-puting power (e.g. spectroscopic measure-ments, modern DCS systems) make on-linetime-varying decisions possible. Along withthese new time-dependent insights, the chemistsin the laboratory start to vary process inputsas a function of time. The potential bene-fit of these additional degrees of freedom isparamount to using optimal control techniques.There are situations where variable input pro-files can be of direct interest:

– There may be a significant theoretical ad-vantage of using a variable profile over thebest constant profile (Rippin, 1983). Theperformance improvement can sometimesbe considerable. In batch crystallization,for example, gains of up to 500% can beobtained by adjusting the temperature, theremoval of solvent or the addition of aprecipitation solvent as functions of time.Large gains are also possible in reactivesemi-batch distillation.

– It is more and more common to adjustthe feed rate in semi-batch reactors so asto force the heat generation to match thecooling capacity of the jacket.

An interesting feature of batch processing is the factthat batch processes are repeated over time. Thus,the operation of the current batch can be improvedby using the off-line measurements available fromprevious batches. The objective is then to get tothe optimum over as few batches as possible. Also,with the tendency to skip pilot plant investigationswhenever possible, this type of process improvementis of considerable interest for the initial batches of anew production campaign.

Implications for Optimization

The industrial situation, as far as technical issues areconcerned, can be summarized as follows:• There is an immediate need to improve the per-

formance of batch processes.• Models are poor, incomplete or nonexistent.• On-line measurements are rare, and state es-

timation is difficult; however, off-line measure-ments can be made available if needed.

• There is considerable uncertainty (model inac-curacies, variations in feedstock, process distur-bances).• Several operational and safety constraints need

to be met.

The implications of the current industrial situationregarding the choice of an appropriate optimizationapproach are presented in Table 1. The details willbe clarified in the forthcoming sections. The mainconclusion is that a framework that uses (preferablyoff-line) measurements rather than a model of theprocess for implementing the optimal inputs is in-deed required.

Industrialsituation

Implications for optimization

Need to improveperformance

Use optimization for comput-ing time-dependent decisions

Absence of a reli-able model

Use measurements for imple-menting optimal inputs

Few on-line mea-surements

Use off-line measurements ina batch-to-batch optimiza-tion scheme

Presence of uncer-tainty

Identify and track signalsthat are invariant to uncer-tainty

Operational andsafety constraints

Track constraints so as to re-duce conservatism

Table 1: Implications of the industrial situation re-garding the choice of an appropriate optimizationapproach

Overview of Batch Process Optimiza-tion

The optimization of batch processes typically in-volves both dynamic and static constraints and fallsunder the class of dynamic optimization. Possiblescenarios in dynamic optimization are depicted inFigure 1. The first level of classification depends onwhether or not uncertainty (e.g., variations in initialconditions, unknown model parameters, or processdisturbances) is considered. The standard approachis to discard uncertainty, leading to a nominal solu-tion that may not even be feasible, let alone optimal,in the presence of uncertainty.

The second level concerns the type of informationthat can be used to combat uncertainty. If mea-surements are not available, a conservative stand is


Dynamic Optimization

Nominal Optimization(discards uncertainty)

Optimization under Uncertainty

No MeasurementsRobust Optimization

(conservative)

MeasurementsMeasurement-based

Optimization

Model-basedRepeated Optimization

Model-freeImplicit Optimization

FixedModel

(accuracyof model)

RefinedModel

(persistencyof excitation)

Evolution/Interpolation(curse of dim-ensionality)

Referencetracking

(what to trackfor optimality)

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Problem:

Uncertainty:

Information:

Input calculation:

Methodology:

Figure 1: Dynamic optimization scenarios with, in parentheses, the corresponding major disadvantage

required. In contrast, conservatism can be reducedwith the use of measurements.

In the next levels, the classification is based on howthe measurements are used in order to guide the op-timization. The calculation of inputs can be eithermodel-based or model-free. In the model-based case,the type of model that is used (fixed or refined) af-fects the resulting methodology. If the input cal-culation is model-free, the current measurement iseither compared to a reference or used for interpo-lation between pre-computed optimal values. Thedifferent scenarios are discussed in detail next.

Nominal Optimization

In nominal optimization, the uncertainty is simplydiscarded. A typical batch optimization problemconsists in achieving a desired product quality atthe most economical cost, or maximizing the prod-uct yield for a given batch time. The optimizationcan be stated mathematically as follows:

minu(t)

J = φ(x(tf )) (1)

s.t. x = F (x, u), x(0) = x0 (2)S(x, u) ≤ 0, T (x(tf )) ≤ 0 (3)

where J is the scalar performance index to be min-imized, x the n-vector of states with known initial

conditions x0, u the m-vector of inputs, F a vec-tor field describing the dynamics of the system, Sthe ζ-vector of path constraints (which include stateconstraints and input bounds), T the τ -vector of ter-minal constraints, φ a smooth scalar function repre-senting the terminal cost, and tf the final time.

The problem (1)–(3) is quite general. Even whenan integral cost needs to be considered, i.e., J =φ(x(tf )) +

∫ tf0

L(x, u)dt, where L a smooth func-tion, it can be converted into the form (1)–(3) bythe introduction of an additional state: xcost =L(x, u), xcost(0) = 0, and J = φ(x(tf ))+xcost(tf ) =φ(x(tf )). Let J∗ be the optimal solution to (1)–(3).It is interesting to note that the minimum time prob-lem with an additional constraint φ(x(tf )) ≤ J∗, i.e,

mintf ,u(t)

tf (4)

s.t. x = F (x, u), x(0) = x0 (5)S(x, u) ≤ 0, T (x(tf )) ≤ 0 (6)φ(x(tf )) ≤ J∗ (7)

will lead to exactly the same optimal inputs as (1)–(3), though the numerical conditioning of the twoproblems, (1)–(3) and (4)–(7), may differ consider-ably. The equivalence of solutions is verified usingthe necessary conditions of optimality. So, withoutloss of generality, the final time will be assumed fixedin this paper.


Application of Pontryagin’s Maximum Principle(PMP) to (1)–(3) results in the following Hamilto-nian (Bryson and Ho, 1975; Kirk, 1970):

H = λT F (x, u) + µT S(x, u) (8)

λT = −∂H

∂x, λT (tf ) =

∂φ

∂x

∣∣∣∣tf

+ νT(∂T

∂x

)∣∣∣∣tf

(9)

where λ(t) �= 0 is the n-vector of adjoint states (La-grange multipliers for the system equations), µ(t) ≥0 the ζ-vector of Lagrange multipliers for the pathconstraints, and ν ≥ 0 the τ -vector of Lagrange mul-tipliers for the terminal constraints. The Lagrangemultipliers µ and ν are nonzero when the corre-sponding constraints are active and zero otherwiseso that µTS(x, u) = 0 and νTT (x(tf )) = 0 always.The first-order necessary condition for optimality ofthe input ui is:

Hui =∂H

∂ui= λT

∂F

∂ui+µT

∂S

∂ui= λTFui+µTSui = 0.

(10)

Note that, in this paper, PMP will not be usedto determine the optimal solution numerically sincethis procedure is well known to be ill-conditioned(Bryson, 1999). However, PMP will be used to de-cipher the characteristics of the optimal solution.

Robust Optimization

In the presence of uncertainty, the classical open-loopimplementation of off-line calculated nominal opti-mal inputs may not lead to optimal performance.Moreover, constraint satisfaction, which becomesimportant in the presence of safety constraints, maynot be guaranteed unless a conservative strategy isadopted. In general, it can be assumed that themodel structure is known and the uncertainty con-centrated in the model parameters and disturbances.Thus, in the uncertain scenario, the optimization canbe formulated as follows:

minu(t)

J = φ(x(tf )) (11)

s.t. x = F (x, θ, u) + d, x(0) = x0

S(x, u) ≤ 0, T (x(tf )) ≤ 0

where θ is the vector of uncertain parameters, andd(t) the unknown disturbance vector. In addition,the initial conditions x0 could also be uncertain.

To solve this optimization problem, an approach re-ferred to as robust optimization, where the uncer-tainty is taken into account explicitly, is proposed

in the literature (Terwiesch et al., 1994). The un-certainty is dealt with by considering all (several)possible values for the uncertain parameters. Theoptimization is performed either by considering theworst-case scenario or in an expected sense. Thesolution obtained is conservative but corresponds tothe best possibility when measurements are not used.

Measurement-based Optimization (MBO)

The conservatism described in the subsection abovecan be considerably reduced with the use of measure-ments, thereby leading to a better cost. Consider theoptimization problem in the presence of uncertaintyand measurements as described below:

minuk[tl,tf ]

Jk = φ(xk(tf )) (12)

s.t. xk = F (xk, θ, uk) + dk, xk(0) = xk0

yk = h(xk, θ) + vk

S(xk, u) ≤ 0, T (xk(tf )) ≤ 0given yj(i), i = {1, · · · , N}

∀ j = {1, · · · , k − 1}, andi = {1, · · · , l} for j = k.

where xk(t) is the state vector, uk(t) the input vec-tor, dk(t) the process disturbance, vk(t) the mea-surement noise, and Jk the cost function for thekth batch. Let y = h(x, θ), a p-dimensional vector,be the combination of states that can be measured,yj(i) the ith measurement taken during the jth

batch, and N the number of measurements within abatch. The objective is to utilize the measurementsfrom the previous (k − 1) batches and the measure-ments up to the current time, tl, of the kth batch inorder to tackle the uncertainty in θ and determinethe optimal input policy for the remaining time in-terval [tl, tf ] of the kth batch.

Role of the model in the calculation of theinputs. Among the measurement-based optimiza-tion schemes, a classification can be done accordingto whether or not a model is used to guide the opti-mization.

Model-based input calculation: Repeated optimiza-tion. In optimization, the model of the system canbe used to predict the evolution of the system, com-pute the cost sensitivity with respect to input vari-ations so as to obtain search directions, and updatethe inputs towards the optimum. Measurements arethen used to estimate the current states and param-eters. As the estimation and optimization tasks are


typically repeated over time, this scheme is often re-ferred to as repeated optimization. The model iseither fixed or refined using measurements, the ad-vantages and disadvantages of which are discussednext.

• Fixed model: If the model is not adjusted, itneeds to be fairly accurate. This, however, isagainst the philosophy of the approach thatassumes the presence of (considerable) uncer-tainty. If the uncertainty is only in the formof disturbances and not in the model parame-ters, it might be sufficient to use a fixed model.On the other hand, if the model is not accu-rate enough, the methodology will have diffi-culty converging to the optimal solution. Notethat, since the measurements are used to esti-mate the states only (and not the parameters),there is no need for persistent inputs.• Refined model: When model refinement is used,

the need to start with an accurate model is al-leviated, but it is necessary to excite appropri-ately the system for estimating the uncertainparameters. However, the optimal inputs maynot provide sufficient excitation. On the otherhand, if sufficiently exciting inputs are providedfor parameter identification, the resulting solu-tion may not be optimal. This leads to a con-flict between the objectives of parameter esti-mation and optimization. This conflict has beenstudied in the adaptive control literature un-der the label dual control problem (Roberts andWilliams, 1981; Wittenmark, 1995).

Model-free input calculation: Implicit optimization.In this approach, measurements are used directly toupdate the inputs towards the optimum, i.e., with-out using a model and explicit numerical optimiza-tion. However, a model might be used a priori tocharacterize the optimal solution. The classificationhere is based on whether the measurement is used forinterpolation between pre-computed optimal valuesor simply compared to a reference.

• Evolution/interpolation: The inputs are com-puted from past data and current measure-ments. If only batch-end measurements areused, the difference in cost between successiveexperimental runs can be used to provide theupdate direction for the inputs (evolutionaryprogramming). The on-line version of this ap-proach is based on the feedback optimal solution– the solution to the Hamilton-Jacobi-Bellmanequation (Kirk, 1970) – being stored in one form

or the other (e.g., neural network, look-up table,or dynamic programming).

The main drawback of this approach is the curseof dimensionality. A large number of experi-mental runs are needed to converge to the opti-mum if only batch-end measurements are used.The use of the method with on-line measure-ments requires either a computationally expen-sive look-up table or a closed-form feedback law,which is analytically expensive or impossible toobtain in many cases.• Reference tracking: The inputs are computed

using feedback controllers that track appropri-ate references. The main question here is whatreferences should be tracked. In most of thestudies found in the literature, the referencescorrespond to optimal trajectories computedoff-line using a nominal model. Such an ap-proach, however, is not guaranteed to be opti-mal in the presence of uncertainty. As will beexplained later, this paper uses the concept ofinvariants to chose references, the tracking ofwhich implies optimality.

Type of measurements. The type of measure-ments (off-line or on-line) can add another level tothe classification of optimization strategies.

• Off-line measurements: Off-line measurementsinclude measurements taken at the end of thebatch (batch-end measurements) and, possibly,off-line analysis of samples taken during thebatch. These measurements cannot be used toimprove the current batch but only subsequentones. Off-line measurements are most commonin industrial practice. They enable the set-upof a batch-to-batch optimization approach to ac-count for parametric uncertainty by exploitingthe fact that batch processes are typically re-peated. Process knowledge obtained from pre-vious batches is used to update the operatingstrategy of the current batch. This approachrequires solving an optimization problem at thebeginning of each batch. The objective is thento get to the optimum over a few batches.• On-line measurements: When information is

available during the course of the batch, an on-line optimization approach can be used. Mea-surements are used to compensate for uncer-tainty both within the batch and, possibly, alsoin a batch-to-batch manner. With this compen-sation, the variability caused by uncertainty isreduced to a large extent. Thus, it is possible


to guarantee feasibility with smaller conserva-tive margins which, in turn, helps improve thecost.

MBO vs. MPC. Model-predictive control (MPC),which has been well studied in the literature (seeRawlings et al. (1994); Qin and Badgwell (1997);Morari and Lee (1999) for surveys), has both someoverlap and differences with MBO schemes that formthe subject of this paper. MPC typically uses therepeated optimization approach to solve a controlproblem in an optimal manner. The major pointsthat distinguish MBO from MPC are discussed next.

• Goal and cost function: The goal in MPC is con-trol – choose the inputs to track given references– whereas the goal in MBO is optimization –maximize the yield of a product, minimize timefor a given productivity, etc. In MPC, the con-trol problem is formulated as an optimizationwith the cost function reflecting the quality ofcontrol, which typically is quadratic in nature,i.e., J =

∫ tf0

(xT Qx + uT Ru

)dt. In contrast,

the cost function in MBO reflects the economicobjective to optimize. However, once the op-timization problem is formulated, similar toolsare used for solution.• Continuous vs. batch processes: MPC is

oriented principally towards continuous pro-cesses. Stability is the main issue there andhas been studied extensively in the MPC lit-erature (Mayne et al., 2000). In contrast, MBOis oriented towards batch processes with a finiteterminal time. Stability does not play a crucialrole, and there is even a tendency to destabi-lize the system towards the end for the sake ofoptimality (the so-called batch kick). The im-portant issues in MBO are feasibility and feed-back optimality – how optimal is the operationin the presence of constraints and uncertainty.In contrast to MPC, MBO schemes can takeadvantage of the fact that batches are typicallyrepeated. Run-to-run and implicit optimizationschemes are thus particular to the MBO litera-ture.• Role of constraints: MBO typically has solu-

tions that are on the constraints since the poten-tial of optimization arises mainly from the pres-ence of path and terminal constraints. Thus, itis important to go as close to the constraintsas possible and, at the same time, guaranteefeasibility. In contrast, though MPC has beendesigned to handle constraints, the typical prob-lems considered in the framework of MPC try

to avoid the solution being on the constraintsby introducing a compromise between trackingperformance and input effort.

Certain MBO schemes in the category of repeatedoptimization have been referred to as MPC schemesin the literature (Eaton and Rawlings, 1990; Mead-ows and Rawlings, 1991; Helbig et al., 1998; Laksh-manan and Arkun, 1999) and, thus, fall in the greyarea between Batch MPC (Lee et al., 1999; Chinet al., 2000) and MBO for batch processes. It mightbe interesting to note that the search for optimal-ity via tracking has also been studied for continu-ous processes. The terms “self-optimizing control”,“feedback control”, or “constraint control” are oftenused. The reader is referred to (Skogestad, 2000) foran overview of the work done in this area.

Classification of measurement-based opti-mization methods. Only MBO methods (as op-posed to MPC methods) that have been designed todeal explicitly with uncertainty in batch processingare considered in the classification. Table 2 illus-trates the interplay between the type of measure-ments (off-line vs. on-line) and the role played bythe model (model-based vs. model-free adaptation).Representative studies available in the literature areplaced in the table.

Invariant-based Optimization

The idea of Invariant-Based Optimization (IBO) isto identify those important characteristics of the op-timal solution that are invariant under uncertaintyand provide them as references to a feedback con-trol scheme. Thus, optimality is achieved by track-ing these references without repeating numerical op-timization. Also, the fact that batches are typi-cally repeated over time can be used advantageously,thereby providing the possibility of on-line and/orbatch-to-batch implementation. The methodologyconsists of:

1. a parsimonious state-dependent parameteriza-tion of the inputs,

2. the choice of signals that are invariant underuncertainty, and

3. the tracking of invariants using measurements.

These three steps are discussed in the following sub-sections.


Methodology Batch-to-batch optimization(Off-line measurements)

On-line optimization(On-line measurements)

Model-basedFixed model

Zafiriou and Zhu (1990)Zafiriou et al. (1995)Dong et al. (1996)

Meadows and Rawlings (1991)Agarwal (1997)Abel et al. (2000)

Model-basedRefined model

Filippi-Bossy et al. (1989)Marchal-Brassely et al. (1992)Rastogi et al. (1992)Fotopoulos et al. (1994)Le Lann et al. (1998)Ge et al. (2000)Martinez (2000)

Eaton and Rawlings (1990)Ruppen et al. (1998)Gattu and Zafiriou (1999)Noda et al. (2000)Lee et al. (2000)

Model-freeEvolutionInterpolation

Clarke-Pringle and MacGregor (1998) Tsen et al. (1996)Rahman and Palanki (1996)Yabuki and MacGregor (1997)Krothapally et al. (1999)Schenker and Agarwal (2000)

Model-freeReferencetracking

Scheid et al. (1999)Srinivasan et al. (2001)

Soroush and Kravaris (1992)Terwiesch and Agarwal (1994)Van Impe and Bastin (1995)Saenz de Buruaga et al. (1997)Ubrich et al. (1999)Fournier et al. (1999)Gentric et al. (1999)Lakshmanan and Arkun (1999)Visser et al. (2000)

Table 2: MBO methods specifically designed to compensate uncertainty

Piecewise Analytic Characterization of theOptimal Solution

The parsimonious state-dependent parameterizationarises from intrinsic characteristics of the optimalsolution. The optimal solution is seen to possess thefollowing properties (Visser et al., 2000):• The inputs are in general discontinuous, but are

analytic in between discontinuities. The timeat which an input switches from one interval toanother is called a switching time.• Two types of arcs (constraint-seeking and

compromise-seeking) are possible betweenswitching instants. In a constraint-seeking arc,the input is determined by a path constraint,while in the other type of interval, the input liesin the interior of the feasible region (Palankiet al., 1993). The set of possible arcs isgenerically labelled η(t).

The structure of the optimal solution is described bythe type and sequence of arcs which can be obtainedin three ways:

• educated guess by an experienced operator,• piecewise analytical expressions for the optimal

inputs,• inspection of the solution obtained from numer-

ical optimization.

In the most common third case, a simplified modelof the process is used to compute a numerical solu-tion in which the various arcs need to be identified.The exercise of obtaining the analytical expressionsfor the optimal inputs is undertaken only if the nu-merical solution cannot be interpreted easily. Thisanalysis is, in general, quite involved and is only in-tended to provide insight into what constitutes theoptimal solution rather than to implement the opti-mal solution. This analysis is discussed next.

Constraint-seeking vs. compromise-seekingarcs.Constraint-seeking arc for ui (Bryson and Ho,1975). In this case, the input ui is determinedby an active constraint, say, Sj(x, u) = 0. Thus,µj �= 0. If Sj(x, u) depends explicitly on ui (e.g., in


the case of input bounds), the computation of the op-timal ui is immediate. Otherwise, since Sj(x, u) = 0over the entire interval under consideration, its timederivatives are also zero, dk

dtkSj(x, u) = 0, for all k.

Note that the time differentiation of Sj(x, u) con-tains x, i.e., the dynamics of the system. Sj(x, u)can be differentiated with respect to time until uiappears. The optimal input is computed from thattime derivative of Sj(x, u) where ui appears. Thecomputed input ui is typically a function of thestates of the system, thus providing a feedback law.

Compromise-seeking arc for ui (Palanki et al.,1993). In this arc, none of the path constraints per-taining to the input ui is active. The input is thendetermined from the necessary condition of optimal-ity, i.e., Hui = λTFui = 0. If Fui depends explic-itly on ui, the computation of the optimal ui is im-mediate. Otherwise, since Hui = 0 over the entireinterval, the time derivatives of Hui are also zero,dk

dtkHui = 0, for all k. Hui can be differentiated with

respect to time until ui appears, from which the op-timal input is computed. The computed input is afunction of the states and might possibly also dependon the adjoint variables. If ui does not appear at allin the time differentiations of Hui , then either nocompromise-seeking arcs exist or the optimal inputui is non-unique (Baumann, 1998).

The fact that the optimal inputs are in the interiorof the feasible region is the mathematical represen-tation of the physical tradeoffs present in the systemand affecting the cost. If there is no intrinsic trade-off, the input ui does not appear in the successivetime differentiations of Hui . This forms an impor-tant subclass for practical applications. It guaran-tees that the optimal solution is always on the pathconstraints. This is the case in controllable linearsystems, feedback-linearizable systems, flat systems,and involutive-accessible systems, a category whichencompasses many practical systems (Palanki et al.,1993; Benthack, 1997).

Constraint-seeking vs. compromise-seekingparameters.Parsimonious input parameterization. The piecesdescribed above are sequenced in an appropriatemanner to obtain the optimal solution. The switch-ing times and, possibly, a few variables that approx-imate the compromise-seeking arcs completely pa-rameterize the inputs. The decision variables of theparameterization are labelled π. In comparison withpiecewise constant or piecewise polynomial approxi-mations, the parameterization proposed is exact and

parsimonious.

Optimal choice of π. Once the inputs have been pa-rameterized as u(π, x, t), the optimization problem(1)–(3) can be written as:

minπ

J = φ(x(tf )) (13)

s.t. x = F (x, u(π, x, t)), x(0) = x0 (14)T (x(tf )) ≤ 0 (15)

Some of the parameters in π are determined byactive terminal constraints (termed the constraint-seeking parameters) and some from sensitivities(termed the compromise-seeking parameters). Notethe similarity with the classification of arcs for inputui. Without loss of generality, let all τ terminal con-straints be active at the optimum. Consequently, thenumber of decision variables arising from the par-simonious parameterization, nπ, needs to be largerthan or equal to τ in order to satisfy all terminalconstraints.

The idea is then to separate those variations in πthat keep the terminal constraints active from thosethat do not affect the terminal constraints. For this,a transformation πT → [πT πT ] is sought such thatπ is a τ -vector and π a (nπ− τ)-vector with ∂T

∂π = 0.A linear transformation which satisfies these proper-ties can always be found in the neighborhood of theoptimum. Then, the necessary conditions for opti-mality of (13)–(15) are:

T = 0,∂φ

∂π+ νT

∂T

∂π= 0, and

∂φ

∂π= 0. (16)

The active constraints T = 0 determine the τdecision variables π while π are determined fromthe sensitivities ∂φ

∂π = 0. Thus, π correspondsto the constraint-seeking parameters and π to thecompromise-seeking parameters. The Lagrange mul-tipliers ν are calculated from ∂φ

∂π + νT ∂T∂π = 0.

Signals Invariant under Uncertainty

In the presence of uncertainty, the numerical val-ues of the optimal input ui in the various arcs andthe input parameters π might change considerably.However, what remains invariant under uncertaintyis the fact that the necessary condition Hui = 0 hasto be verified. Hui = 0 takes on different expres-sions for constraint-seeking and compromise-seekingarcs. To ease the development, it is assumed that theuncertainty is such that it does not affect the type


and sequence of arcs nor the set of active terminalconstraints.

Choice of invariants.Choice of invariants for constraint-seeking andcompromise-seeking arcs. A set of signals Iηi (t) =hηi (x(t), u(t), t), referred to as invariants for arcs,will be chosen such that optimality is achieved bytracking Iηref,i = 0. Note the dependence of hηi withrespect to t, which indicates that hηi can be differentin different intervals of the optimal solution.

Tracking Hui = 0 has different interpretations withrespect to the two types of arcs. In the case ofa constraint-seeking arc for ui with the constraintSj being active, Hui = λTFui + µj

∂Sj∂ui

= 0, withµj �= 0 and λTFui �= 0. The constraint has to beactive for the sake of optimality since otherwise µjis zero and Hui �= 0. Thus, the invariant along aconstraint-seeking arc is hηi (x, u, t) = Sj(x, u). For acompromise-seeking arc, Hui = λTFui = 0. There-fore, the invariant is hηi (x, u, t) = λTFui(x, u).

Note that the element that remains invariant despiteuncertainty is the fact that optimal operation cor-responds to Iηref = 0. However, the uncertaintydoes have an influence on the value of Iη(t), andthe inputs need to be adapted in order to guaranteeIηref = 0.

Choice of invariants for constraint-seeking andcompromise-seeking parameters. In addition to thechoice of invariants for the various arcs, it is im-portant to choose the invariants for the parame-ters π. Following similar arguments, a set of sig-nals Iπ = hπ(x(tf )) can be constructed such thatthe optimum corresponds to Iπref = 0, also in thepresence of uncertainty. Clearly, the invariantsarise from the conditions of optimality. For theconstraint-seeking parameters, they correspond tothe terminal constraints hπ(x(tf )) = T (x(tf )) and,for the compromise-seeking parameters, to sensitiv-ities hπ(x(tf )) = ∂φ(x(tf ))

∂π .

To summarize, the invariants are as follows:

• For constraint-seeking arcs:hηi (x, u, t) = Sj(x, u)

• For compromise-seeking arcs:hηi (x, u, t) = λTFui(x, u)• For constraint-seeking parameters:hπ(x(tf )) = T (x(tf ))• For compromise-seeking parameters:hπ(x(tf )) = ∂φ(x(tf ))

∂π

Sensitivity of the cost. The sensitivity of thecost to non-optimal operation is in general muchlower along a compromise-seeking arc than alonga constraint-seeking arc. Consider the optimal in-put ui determined by the path constraint Sj andthe optimality condition Hui = λTFui + µj

∂Sj∂ui

= 0with µj �= 0. If the input does not keep the con-straint active, µj becomes zero. Thus, the change incost is directly proportional to λTFui , which is non-zero. In contrast, along a compromise-seeking arc,Hui = λTFui = 0, and a small deviation of ui fromthe optimal trajectory will result in a negligible lossin cost. Similarly, as seen from (16), the deviation incost arising from the non-satisfaction of a terminalconstraint is proportional to νT ∂T

∂π , whilst a smallvariation of π cause negligible loss in cost.

In summary, it is far more important to track thepath constraints Sj than the sensitivity λTFui . Fur-thermore, it is far more important to track the ter-minal constraints T than the sensitivities ∂φ

∂π . Conse-quently, it is often sufficient in practical situations tofocus attention only on constraint-seeking arcs andparameters.

Tracking of Invariants

The core idea of the optimization scheme is to usea model to determine the structure of the optimalinputs and measurements to update a few input pa-rameters and the value of the inputs in some of theintervals. This way, the optimal inputs are deter-mined directly from process measurements and notfrom a (possibly inaccurate) model.

Optimality despite uncertainty is approached byworking close to the active constraints, i.e., wherethere is much to gain! Indeed, tracking path andterminal constraints is usually much more importantthan regulating sensitivities as was argued above.The structure given in Figure 2 is proposed to trackthe invariants by use of feedback (Srinivasan et al.,1997; Visser, 1999; Visser et al., 2000). The two ma-jor blocks are described below:• Analysis: This level consists of a simplified (not

necessarily accurate) model of the process. Thetasks are as follows:

1. The numerical optimizer solves the op-timization problem using the simplifiedmodel and provides the nominal signals x∗

and u∗.

2. The type and sequence of arcs are deci-


ProcessTrajectory

GeneratorPath

Controller

Terminal

Controller

Construction

of Invariants

On-line

Off-line

Characterization

of the solution

Choice of

Invariants

ANALYSIS

OPTIMALITY THROUGH FEEDBACK

Numerical

Optimizer

Simplified

Model

π

η

IηIπ

00

−−

u y

d

x∗

u∗hη , hπ

π∗

η∗

Figure 2: Invariant-based optimization

phered from the numerical solution, lead-ing to a parsimonious parameterization ofthe inputs (characterization of the optimalsolution).

3. The invariant functions hη(x(t), u(t), t)and hπ(x(tf )) are obtained as proposedearlier. Note that the switching strategyis inherent in the choice of hη.

• Optimality through feedback: The invariantsIηref = 0 and Iπref = 0 are tracked with the helpof path and terminal feedback controllers, re-spectively. The trajectory generator computesthe current inputs as a function of η(t) and π.

The model-based ‘Analysis’ is normally carried outoff-line once, and only the measurement-based ‘Op-timality through feedback’ operates on-line duringprocess runs. Thus, the implementation is model-free and measurement-based.

Practical applicability of IBO

If the model and the true (unknown) system sharethe same input structure (type and sequence of arcs)and the same active terminal constraints, IBO willbe capable of optimizing the true system. Thus,the value of IBO in practice will depend on boththe robustness of the proposed input structure with

respect to uncertainty (modeling errors and distur-bances) and the ability to measure the path and ter-minal constraints. These issues are briefly discussednext.

• Role of the model. Although the implementa-tion of optimal inputs in the invariant-basedscheme is truly model-free, a model may stillbe needed at the analysis step. The role of themodel is to determine the structure of the op-timal inputs, i.e., the type and the sequence ofarcs and the set of active constraints. The struc-ture of the inputs is obtained either numericallyor via educated guesses, with the proposed in-put structure being verified using PMP neces-sary conditions on the nominal model. The ap-proach is directly applicable to large-scale in-dustrial process, as long as the nominal modeland the real system share the same input struc-ture. So, in contrast to model-based approachesor what is sought of for simulation purposes,there is no need for a detailed model or for accu-rate parameter values. The model simply needsto reflect the major tradeoffs specific to the op-timization problem at hand. The parts of themodel that do not address these effects can bediscarded.• Construction of invariants from measurements.

Since Iη and Iπ are not measured directly,they need to be reconstructed from the avail-able measurements. In the case of constraint-


seeking arcs and parameters, the invariants cor-respond to physical quantities (path or terminalconstraints). Off-line measurements of terminalquantities are in general available. On-line mea-surements to meet path constraints might bemore difficult. Three cases can be considered:

1. The path constraint deals directly with aphysical quantity that can be easily mea-sured such as temperature or pressure.

2. The path constraint deals with a quantitythat cannot be directly measured, but theconstraint can be rewritten with respect tosomething that can be measured. This is,for example, possible when a heat removalconstraint can be rewritten as a constrainton a cooling temperature.

3. Cases 1 and 2 do not apply, and some typeof inference or state estimation is necessaryto meet the constraint. This case is clearlymore involved than the two preceding ones.The reconstruction problem is closely re-lated to inferential control (Joseph andBrosilow, 1978; Doyle, 1998). However,it may well happen that a conservativeapproach for meeting the path constraint(requiring easily-available or no measure-ments) is sufficient.

On the other hand, for compromise-seeking arcsand parameters, the invariants are sensitivities.For computation of sensitivities, either a modelof the process or multiple process runs are re-quired, which is typically more difficult. How-ever, as discussed above, the sensitivity with re-spect to input variations in compromise-seekingarcs and parameters can often be neglected. Insuch a case, all compromise-seeking arcs and pa-rameters are kept at their off-line determinedvalues, and only the constraint-seeking arcs andparameters are adjusted.

• Difference in time scale – on-line vs. off-linemeasurements. In general, there is a differencein time scale between the path controller andthe terminal controller. The path controllerworks within a batch using on-line measure-ments (running index is the batch time t) (Ben-thack, 1997). The terminal controller operateson a batch-to-batch basis using off-line measure-ments (running index is the batch number k)(Srinivasan et al., 2001).

If on-line measurements are not available, thepath controller is inactive. If off-line measure-ments of the path constraint are available, it is

possible to use the path controller in a batch-to-batch mode so that the system will be closerto the path constraint during the next batch(Moore, 1993). On the other hand, if it is pos-sible to predict Iπ from on-line measurements,it might be possible to use the terminal con-troller within the batch (Yabuki and MacGre-gor, 1997).

The presence of disturbances influences bothη(t) and π. Disturbances affecting η(t) withinthe batch are rejected by the path controller.However, the effect of any disturbance withinthe batch on π cannot be rejected since the ter-minal controller only works on a batch-to-batchbasis. Constant disturbances (e.g. raw materialvariations) can be rejected from batch-to-batchby the terminal controller.

• Backoff from constraints. In the presence of dis-turbances and parametric uncertainty that can-not be compensated for by feedback, the useof conservative margins, called backoffs, is in-evitable to ensure feasibility of the optimizationproblem (Visser et al., 2000). The presence ofmeasurement errors also necessitates a backoff.Based on an estimate of the uncertainty, theprobability density function of the state vari-ables can be calculated. The margins are thenchosen such that the spread of the states re-mains within the feasible region with a certainconfidence level. Note that the margins typi-cally vary with time.

• Reduction of backoff. Due to the sensitivity re-duction characteristic of feedback control, theconservatism can be reduced considerably inthe proposed framework in comparison withthe standard open-loop optimization schemes.The feedback parameters can be chosen so asto minimize the spread in the state variablesresulting from uncertainty. The use of feed-back becomes particularly important when theuncertainty tends to increase during a batchrun. With reduced backoffs, the process canbe driven closer to active constraints, therebyleading to improved performance.

Example - Semi-batch Reactor withSafety and Selectivity Constraints

Description of the Reaction System

• Reaction: A + B → C, 2 B → D.

• Conditions: Semi-batch, isothermal.


• Objective: Maximize the amount of C at a givenfinal time.• Manipulated variable: Feed rate of B.• Constraints: Input bounds; limitation on the

heat removal rate through the jacket; constrainton the amount of D produced.• Comments: The reactor is kept isothermal by

(say) adjusting the cooling temperature in thejacket, Tc. B is fed at the temperature Tin = T .To remain isothermal, the power generated bythe reactions, qrx, must be evacuated throughthe cooling jacket, i.e., qrx = UA(T−Tc). Thus,the heat removal constraint can be expressed interms of a lower bound for the cooling temper-ature, Tcmin. The variables and parameters aredescribed in the next subsection.Without any constraints, optimal operationwould consist of adding the available B at ini-tial time (i.e., batch mode). The presence of theheat removal constraints calls for semi-batch op-eration with constraint-seeking arcs. Further-more, the constraint on the amount of D thatcan be produced imposes a compromise-seekingfeeding of B in order to maximize C withoutviolating the terminal constraint on D.

Problem Formulation

Variables and parameters: cX : Concentration ofspecies X, nX : Number of moles of species X, V :Reactor volume, u: Feed rate of B, cBin: Inlet con-centration of B, k1, k2: Kinetic parameters, ∆H1,∆H2: Reaction enthalpies, T : Reactor temperature,Tc: Cooling temperature in the jacket, Tin: Feedtemperature, U : Heat transfer coefficient, A: Sur-face for heat transfer, and qrx: power produced bythe reactions. The numerical values are given in Ta-ble 3.

Model equations:

˙cA = −k1cAcB −u

VcA (17)

˙cB = −k1cAcB − 2 k2c2B +

u

V(cBin − cB)(18)

V = u (19)

with the initial conditions cA(0) = cAo, cB(0) =cBo, and V = Vo. Assuming cC(0) = cD(0) =0, the concentration of the species C and Dare given by cC = cAoVo−cAV

V and cD =(cBoVo−cBV )+cBin(V−Vo)−(cAoVo−cAV )

2V . The powerproduced by the reactions and Tc are given by

qrx = k1 cA cB(−∆H1)V + 2 k2 c2B(−∆H2)V (20)

Tc = T − qrxU A

(21)

Optimization problem:

maxu(t)

J = nC(tf ) (22)

s.t. (17)− (19)Tc(t) ≥ Tcmin

nD(tf ) ≤ nDfmax

umin ≤ u ≤ umax

k1 0.75 l/(mol h)k2 0.014 l/(mol h)∆H1 −7× 104 J/mol∆H2 −5× 104 J/molcBin 10 mol/lUA 8× 105 J/Kh

umin 0 l/humax 100 l/hTcmin 10 oCnDfmax 5 mol

cAo 2 mol/lcBo 0 mol/lVo 500 ltf 2.5 h

Table 3: Model parameters, operating bounds andinitial conditions

Piecewise Analytic Characterization

Using Pontryagin’s Maximum Principle, it can beshown that the competition between the two re-actions results in a feed that reflects the optimalcompromise between producing C and D. Thiscompromise-seeking input can be calculated fromthe second time derivative of Hu as:

ucomp =V cB(k1cA(2 cBin − cB) + 4 k2cBcBin)

2 cBin (cBin − cB)(23)

The other possible arcs correspond to the input be-ing determined by the constraints: (i) u = umin,(ii) u = umax, and (iii) u = upath. The inputupath corresponds to riding along the path constraintTc = Tcmin. The input is obtained by differentiatingthe path constraint once with respect to time, i.e.,from Tc = 0 :

upath =ND

∣∣∣∣Tc=Tcmin

(24)

N = cBV(∆H1k1cA(k1(cA + cB) + 2 k2cB)

+ 4 ∆H2k2cB(k1cA + 2 k2cB))


D = ∆H1k1cA(cBin − cB)+ 4 ∆H2k2cB(2 cBin − cB)

Sequence of arcs and parsimonious parameteriza-tion:

• The input is initially at the upper bound, u =umax, in order to attain the path constraint asquickly as possible.• Once Tc reaches Tcmin, u = upath is applied in

order to keep Tc = Tcmin.• The input switches to u = ucomp at the time

instant π so as to take advantage of the optimalcompromise in order to maximize nc(tf ) andmeet the terminal constraint nD(tf ) = nDfmax .

Since analytical expressions for the input in thevarious arcs exist, the optimal solution can be pa-rameterized using only the switching time betweenthe path constraint and the compromise-seeking arc.This parameter π is determined numerically so asto satisfy the terminal constraint nD(tf ) = nDfmax .The invariants Iη correspond to the input boundin the first interval, the path constraint in thesecond interval and the sensitivity λTFu for thecompromise-seeking arc. For the switching time, theinvariant is the terminal constraint itself. The op-timal input is shown in Figure 3 with the optimalvalues π = 1.31h and J = 600.6 mol.

Note that the input upath given by (24) will keep thesystem on the path constraint once the path con-straint Tc = Tcmin is attained, but will not keep thepath constraint active in the presence of uncertainty.The same can be said for ucomp in (23). Thus, theanalytical expressions for upath and ucomp will onlybe used for interpretation of the nominal optimaltrajectory and not for implementation of the trueoptimal solution. At the implementation level, sim-ple PI-controllers will be used.

Measurement-based Optimization

In practice, there can be considerable uncertaintyboth in the stoichiometric and kinetic models. Thisis reflected here as some uncertainty for the kineticparameter k1 in the interval 0.4 ≤ k1 ≤ 1.2 (Thenominal value k1 = 0.75 used in the simulationis assumed to be unknown). In order not to vio-late the constraints, a conservative feed profile (Fig-ure 3) would have to be designed so that: i) thepath constraint is not violated for k1 = k1max = 1.2,

0 0.5 1 1.5 2 2.510

20

30

40

50

60

70

80

90

100Input

Time [h]

u [l/h]

π

umax

upath

ucomp

Figure 3: Nominal optimal input (solid) and Con-servative optimal input (dotted)

and ii) the terminal constraint is not violated fork1 = k1min = 0.4 (a small k1 corresponds to moreB in the reactor and thus to a higher productionof D). So, the conservative profile would consist ofcomputing upath and the first switching instant us-ing k1 = k1max, and adjusting π so that the terminalconstraint is satisfied for k1 = k1min.

With respect to the measurements available, differ-ent optimization scenarios are considered:

1. No measurements: The conservative optimalfeed profile defined above is applied open loopto the simulated nominal plant.

2. Batch-end measurements: Only the measure-ment of nD(tf ) is available and, thus, theswitching time π is updated in a batch-to-batchmanner. For the second interval, upath = uconspath,the conservative value computed off-line usingk1max is applied. Due to the low sensitivity ofthe cost with respect to the fine shape of theinput in the compromise-seeking interval, thelatter is approximated by the constant valueucomp = 20 (l/h).

3. On-line and batch-end measurements: On-linemeasurement of the cooling jacket temperatureTc is available. The path constraint is keptactive using the feedback upath(t) = uconspath +kp (Tcmin−Tc(t))+ki

∫ t

0(Tcmin−Tc(t))dt, where

kp and ki are the parameters of a PI controller.In addition, the switching time π is updatedin a batch-to-batch manner. As in Scenario 2,the compromise-seeking arc is approximated byucomp = 20 (l/h).

The cases of both noise-free and noisy measurements


Terminal Constraint Path ConstraintOptimization Scenario nD(tf ) mol min

tTc(t) oC Cost Loss

(nDfmax = 5 mol) (Tc,min = 10oC) (mol)Open-loop application

1 of optimal 2.71 12.87 498.8 20%conservative inputAdaptation of πusing off-line 4.75 11.62 582.6 3%measurements (with 5% noise)

2 Adaptation of πusing off-line 5.00 11.50 589.2 2%measurements (no noise)Adaptation of upath(t) and πusing on-line and off-line 4.75 11.25 590.9 1.5%measurements (with 5% noise)

3 Adaptation of upath(t) and πusing on-line and off-line 5.00 10.00 600.5 0.02%measurements (no noise)

Table 4: Invariant-based optimization. Results averaged over 100 noise realizations, each consisting ofrun-to-run adaptation over 50 batches.

(5% relative Gaussian measurement noise) are con-sidered. The results are given in Table 4. If the mea-surements are noisy, a conservative margin (backoff)needs to be incorporated so as to guarantee feasi-bility. The backoffs are 0.25 mol for nDfmax and1.25oC for Tcmin.

It is seen that with only off-line (or batch-end) mea-surements, the terminal constraint can be satisfiedby adapting the switching instant π. The evolutionsof the switching instant and the cost for batch-to-batch optimization are shown in Figures 4 and 5.It can be seen that the solution gets close to theoptimum within a few batches.

If, in addition, on-line measurements are available,the path constraint can be kept active as well. Thus,it is possible to get very close to the optimum byusing measurements. The loss of 0.02% in the lastnoise-free scenario is due to the approximation ofthe compromise-seeking arc by the constant valueucomp = 20 (l/h).

Discussion

The model was only necessary to obtain the type andsequence of arcs: umax, upath and ucomp. As far asthe implementation is concerned, umax = 100 l/h ispart of the problem formulation, upath is determined

0 10 20 30 40 501

1.5

2

Evolution of Switching time

Batch Number

π [h]

Figure 4: Evolution of switching time for one real-ization of the batch-to-batch optimization with onlybatch-end measurements (5% measurement noise)

by a PI-controller upon tracking Tcmin, and ucomp

= 20 l/h is a constant-value approximation to theoptimal profile computed off-line using the model.The switching time π between upath and ucomp isadjusted in a run-to-run manner by a PI-controllerin order to meet nDfmax . The actual value of ucomp

is of little relevance as any error in ucomp can beeasily compensated for by an appropriate shift in π.

Assume that, in addition to the two modeled reac-tions, the true system also includes B + C → E,


0 10 20 30 40 50490

500

510

520

530

540

550

560

570

580

590Evolution of Cost

Batch Number

J [mol]

Figure 5: Evolution of cost for the same scenario asin Figure 4

B → F . This would not affect the type and sequenceof arcs since the two additional reactions are similarto the second reaction with respect to the effect ofthe input u, i.e., they consume B away from the first(desired) reaction. Thus the proposed scheme wouldbe equally applicable even in the presence of the twoadditional reactions.

In the formulation of the optimization problem itwas assumed Tin = T . Even without this assump-tion, the proposed approach is applicable. The possi-bility of removing heat through temperature increaseof the feed from Tin to T (so-called sensible heat)changes the heat removal constraint to:

qrx − qin ≤ UA(T − Tcmin) (25)

with qin the rate of heat removal due to the feed ofB. However, the implementation of the heat removalconstraint remains unchanged as it concerns only theRHS of (25): upath(t) is determined as the output ofa PI-controller designed to track Tcmin.

A final important remark: The model parametersgiven in Table 3 are not used for calculating theoptimal feed rate. Only the off-line measurementof nD(tf ) and the on-line measurement of Tc(t) areused to implement the proposed optimizing scheme.

Conclusions

This paper has addressed several optimization issuesthat directly affect the operation of batch processes.It is argued that process improvement is necessaryfor the economic well-being of many batch manufac-

turers. The industrial practice specific to the batchspecialty chemistry is presented, with an emphasison both organizational and technical problems. Onthe organizational side, the lack of global thinking indealing with the individual steps of a complex pro-cess limits the potential for performance improve-ment. On the technical side, important limitationsregarding both modeling and measurement aspectsimpair the use of optimization techniques. In ad-dition, batch processes are characterized by a con-siderable amount of uncertainty and the presence ofoperational and safety constraints.

The lack of reliable models, together with the pres-ence of uncertainty, has favored the investigationof process improvement via utilization of measure-ments (sometimes on-line, most often off-line). Thispaper has classified measurement-based optimiza-tion methods reported in the literature according towhether or not a model is used to guide the optimiza-tion and the type of measurements (on-line, off-line)available.

The major contribution towards process improve-ment of a constrained batch process is through oper-ation on active constraints. Thus, a feedback-basedframework has been proposed to keep the system‘close’ to the active constraints. If only off-line mea-surements are available, this framework results in abatch-to batch optimization scheme with the objec-tive to meet the terminal constraints within a fewbatches. If on-line measurements are available, thepath constraints can also be kept active.

The proposed invariant-based optimization schemeaddresses most of the requirements stemming fromindustrial practice and needs that were listed in Ta-ble 1. More specifically,

• it is aimed at process improvement via the useof time-dependent inputs,

• it is model-independent as far as implementa-tion is concerned,

• if necessary, it uses only available off-line mea-surements,

• it is robust against uncertainty since signals thatare invariant under uncertainty are tracked, andfinally,

• it guarantees feasibility since the constraints areapproached from the safe side.

The approach proposed is effective when the opti-mization potential stems mainly from meeting path


and/or terminal constraints. Such is the case in mostof the batch process optimization problems.

It is possible to perceive the proposed feedback-based optimization strategy from an industrial per-spective. Classical PID control is the most popu-lar technique used currently in industry, and trad-ing it to attain optimality is unacceptable industri-ally. Therefore, in contrast to most model-based op-timization studies, this work attempts to use feed-back control for the sake of optimality. In this sense,the approach has great industrial potential and couldhelp take optimization to the batch chemical indus-try.

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