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Journal of Process Control 17 (2007) 229–240

Distributional uncertainty analysis using power seriesand polynomial chaos expansions

Z.K. Nagy a,*, R.D. Braatz b

a Loughborough University, Department of Chemical Engineering, Loughborough, Leics LE11 3TU, United Kingdomb Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue,

Urbana, IL 6180, United States

Received 23 July 2006; accepted 3 October 2006

Abstract

This paper provides an overview of computationally efficient approaches for quantifying the influence of parameter uncertainties onthe states and outputs of nonlinear dynamical systems with finite-time control trajectories, focusing primarily on computing probabilitydistributions. The advantages and disadvantages of various uncertainty analysis approaches, which use approximate representations ofthe full nonlinear model using power series or polynomial chaos expansions, are discussed in terms of computational cost and accuracy incomputing the shape and tails of the state and output distributions. Application of the uncertainty analysis methods to a simulationstudy is used to provide advice as to which uncertainty analysis methods to select for a particular application. In particular, the resultsindicate that first-order series analysis can be accurate enough for the design of real-time robust feedback controllers for batch processes,although it is cautioned that the accuracy of such analysis should be confirmed a posteriori using a more accurate uncertainty analysismethod. The polynomial chaos expansion is well suited to robust design and control when the objectives are strongly dependent on theshape or tails of the distributions of product quality or economic objectives.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Distributional robustness analysis; Polynomial chaos expansion; Robust control; Nonlinear control; Optimal control

1. Introduction

A comprehensive uncertainty analysis of mechanisticmodels is crucial, especially when these models are usedin the optimal control of processes, which normally occursclose to safety and performance constraints. The model-based computation of optimal control policies for batchand semibatch processes is of increasing importance dueto the industrial interest in increasing productivity in thespecialty chemicals, polymers, pharmaceuticals, and otherindustries [1–3]. However, uncertainty almost always existsin chemical systems and its disregard can lead to the lossof all of the benefits of using optimal control [4]. This

0959-1524/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2006.10.008

* Corresponding author. Tel.: +44 1509 222516; fax: +44 1509 223923.E-mail address: z.k.nagy@lboro.ac.uk (Z.K. Nagy).

motivates the development of techniques to quantify theinfluence of parameter uncertainties on the process statesand outputs, or take uncertainties into account in thedesign of processes or control systems [5]. There is anincreasing need to develop generic formalisms that facili-tate the quantification of uncertainty and its effect onmodel-based predictions. Quantitative estimates obtainedfrom robustness analysis can be used to evaluate the prob-ability of extreme events such as failures, to estimate theconfidence in model predictions, or to design efficientschemes for model or data refinement, for example bydeciding whether more laboratory experiments are neededto provide better parameter [6,7]. Uncertainty propagationhas a key role in robust optimization where appropriateback-off terms, which guarantee that important constraints,such as those that arise from safety or environmental con-siderations, are satisfied under uncertain conditions.

230 Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240

Uncertainty analysis consists of the characterization ofuncertainty in model parameters or inputs based on theirprobability density functions (pdfs) and then propagatingthese pdfs through the model equations to obtain the pdfsof the model outputs. A number of techniques developedby various communities (including climate modelling, hur-ricane prediction, and batch process control) are availablefor uncertainty propagation. The propagation of uncer-tainties via traditional Monte Carlo methods based onstandard or Latin Hypercube sampling is valid for a widerange of problems, however, it is very computationallyexpensive since it requires a large number of simulationsusing the full model. Although this caveat is becoming lesssignificant for analysis purposes, it still can be prohibitive ifthe propagation has to be performed in real-time, such asin robust optimizing control. Another more intrinsic disad-vantage of the classical Monte Carlo method is that it doesnot provide a manageable representation of the predictedprocess. Therefore there is a need to investigate computa-tionally-efficient alternative techniques for uncertaintypropagation, which also provide a simplified mathematicalrepresentation of the process. One alternative approach tothe Monte Carlo procedure is the perturbation method, orsensitivity analysis [8]. This method represents the predic-tion as a perturbation around its nominal value, usuallyassociated with the mean or expected value of the uncertaindeterministic problem. Sensitivity analysis usually providesreliable predictions only when the associated perturbationsare small, and it is not used to predict the shape of thewhole distribution. An alternative way of considering theuncertainty propagation problem is based on probabilitytheory, and consists of representing the random variablesand model in terms of convergent expansions accordingto a framework, which is similar to the deterministicapproximation approaches.

The paper investigates computationally-efficient meth-ods for estimating the entire distribution of states and out-puts of generic batch processes. One class of methods isbased on first- and second-order power series [9] and canbe considered as a combination of the sensitivity analysisand probability theory techniques. The first-order methodis a very computationally efficient way to propagate uncer-tainties but does not capture the shape and tails of nonG-aussian distributions; for strongly nonlinear processes thenumber of terms in the power series expansion has to beincreased, requiring higher computational cost. An alterna-tive approach, belonging to the category of probability the-ory methods, is based on the stochastic response surfaceapproach (SRS), which models the performance function/model output, as a sum of elementary functions (bases)of stochastic input parameters. The efficient applicationof the SRS approach requires the proper choice of the ele-mentary functions and the number and location of the sam-pling points. This paper will consider a variant of the SRSapproach based on the polynomial chaos expansion (PCE)[10–13], which uses Hermite polynomial bases and providesa closed-form functional approximation of the mathemati-

cal model from a significantly lower number of model sim-ulations than those required by conventional methods. Thepower series and polynomial chaos expansion are suitablefor studying the uncertainty propagation in open-loop orclosed-loop systems, and can be applied for uncertaintiesin model parameters, initial conditions, or inputs. The tech-niques are compared with the Monte Carlo method andapplied to compute the distributions of the states and out-puts for the batch crystallization of an inorganic chemicalsubject to uncertainties in the nucleation and growth kinet-ics. The simulation study is used to draw some more gen-eral recommendations about which uncertainty analysismethods to select for a particular application.

2. Distributional uncertainty analysis

2.1. Uncertainty description

Consider the class of finite time (batch) processesdescribed by the generic ODE:

_xðtÞ ¼ f ðxðtÞ; uðtÞ; hÞ ð1Þwith the state vector x 2 Rnx with initial values x0, the vec-tor of control inputs u 2 Rnu , the uncertain parameter vec-tor h 2 Rnh , and the vector function f which is continuouswith respect to its elements. This paper considers the quan-tification of the effect of parametric uncertainties on theaccuracy of the model predictions, which can be used to as-sess whether more experimental data are needed to furtherrefine the parameters. It is assumed that model develop-ment is detailed enough to capture the main structuralcharacteristics of the system, so that uncertainties in themodel structure can be neglected.

Uncertainty quantification can be decomposed in threemain steps: (i) characterisation of the parameter uncertainty,(ii) propagation of uncertainty, and (iii) uncertainty man-agement or evaluation and exploitation of the informationobtained. The uncertainties in the parameters can be repre-sented in terms of standard normal variables or hard boundson the parameters. Define h as the nominal model parametervector of dimension (n · 1), dh as the perturbation about h,and the model parameter vector for the real system as

h ¼ hþ dh: ð2Þ

Hard bounds on the uncertainty in the parameter ischaracterized by the generalized ball

P , h 2 Rnh h� h��� ��� 6 1��� on

ð3Þ

defined by using an appropriate norm k Æ k in Rnh such as ascaled Holder p-norm (1 6 p 61), given by khk ¼kW�1

h hkp, with the invertible weighting matrixWh 2 Rnh�nh . This generalized description of the uncertaintyset includes the case of a confidence hyper-ellipsoid

eh ¼ h : h� h� �T

V�1h h� h� �

6 r2ðaÞ� �

, [14], for a Gauss-

ian random variable vector h with expected value eðhÞ ¼ h,the (nh · nh) positive-definite variance–covariance matrix

Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240 231

Vh, and the scalar r which is the chi-squared distributionfunction with nh degrees of freedom (v2

nhðaÞ) for a chosen

confidence level a:

PellipsoidðaÞ , h 2 Rnh ð1=rðaÞÞV�1=2h h� h� ���� ���

26 1

��� on:

ð4ÞThe generalized ball described by (2) also includes the caseof known lower and upper bounds hl, hu on the uncertainparameters, leading to an uncertainty hyper-box:

Pbox , h 2 Rnh jhl 6 h 6 huf g

¼ h 2 Rnh1

2diagðhu � hlÞ h� 1

2ðhu þ hlÞ

� ��������16 1

����� �

:

ð5ÞAlthough the focus on this manuscript is on uncertain-

ties in parameters, any generic method for analyzing uncer-tainties in parameters can be extended, with minormodifications, to analyze the effects of simultaneous uncer-tainties in parameters, initial conditions, and inputs (as dis-cussed by [6]).

2.2. Power series expansion based approaches for

worst-case and distributional robustness analysis

Define w as a state or output of a process with the nom-inal model parameters h, w as its value for the perturbedmodel parameter vector h, and the difference dw ¼ w� w.The worst-case robustness approach [4,9] writes dw as apower series in dh:

dw ¼ Ldhþ 1

2dhTMdhþ � � � ; ð6Þ

where the jacobian L 2 Rnh , and hessian M 2 Rnh�nh are:

LðtÞ ¼ owðtÞoh

� h

; ð7Þ

MðtÞ ¼ o2wðtÞoh2

� h

: ð8Þ

The elements of the time-varying sensitivity vector L(t) andmatrix M(t) can be computed using finite differences or byintegrating the model’s differential–algebraic equationsaugmented with an additional set of differential equationsknown as sensitivity equations [15]

_L ¼ JxLþ Jh ð9Þwith the matrixes Jx ¼ df =dx 2 Rn2�nx and Jh ¼ df =dh 2Rnx�nh .

When a first-order series expansion is used, analyticalexpressions of the worst-case deviation in the performanceindex (dww.c.) can be computed and the analysis can be per-formed with low computational cost [16]. In the case of anellipsoidal uncertainty description, the worst-case deviationis defined by

oww:c:ðtÞ ¼ maxkWhdhk61

jLðtÞdhj ð10Þ

which has an analytical solution

oww:c:ðtÞ ¼ rðaÞLðtÞVhLTðtÞ �1=2

: ð11Þ

ohw:c:ðtÞ ¼ðrðaÞÞ1=2

LðtÞVhLTðtÞ �1=2

VhLTðtÞ: ð12Þ

A probability density function (pdf) for the model param-eters is needed to compute the pdf of the performance in-dex. More than 90% of the available algorithms toestimate parameters from experimental data [14] producea multivariate normal distribution:

fp:d:ðhÞ ¼1

ð2pÞnh=2 detðVhÞ1=2exp

1

2h� h� �T

V�1h h� h� �� �

:

ð13Þ

When a first-order series expansion is used to relate dw anddh, then the estimated pdf of w is

fp:d:ðwÞ ¼1

V 1=2w

ffiffiffiffiffiffi2pp exp � w� w

� �2

=ð2V wÞ�

; ð14Þ

where the variance of w is

V w ¼ LVhLT: ð15Þ

The distribution is a function of time since the nominalvalue for w and the vector of sensitivities L are functionsof time.

2.2.1. Propagation of probability distribution using the

Monte Carlo method or by contour mappingIn the Monte Carlo method for uncertainty analysis, a

large number of parameters are generated based on themultidimensional probability distribution function of theparameter space and are propagated through the nonlinearmodel, and the probability distribution functions of theperformance index or model states/outputs are constructedfrom the histograms resulted from the nonlinear dynamicsimulations. This approach is illustrated schematically inFig. 1 for a system with three uncertain parameters inwhich the uncertainty is given by a multivariate normal dis-tribution with confidence hyper-ellipsoid shown. Points aretaken randomly form the hyper-ellipsoid and propagatedthrough the nonlinear model, which can be any system ofordinary differential equations (ODE), differential alge-braic equations (DAE), partial differential equations(PDE), or the combination of such equations. The maindisadvantage of the classical Monte Carlo techniques isthat an accurate construction of the output distributionrequires a very large number of parameter samples andcomputer-intensive simulations, to accurately model thetails of the distribution, which correspond to parameterswith low probability of occurrence. In the case illustratedin Fig. 1, although 10,000 sets of parameters were simula-tion, a significant portion of the parameter ellipsoid wasnot well sampled, leading to errors in representing the tailsof the output distribution. Although these parameter

Fig. 1. Schematic representation of the uncertainty propagation through the full nonlinear model using the Monte Carlo method.

232 Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240

values have a low probability to occur, such parametersmay have significant effects on the performance index, withlarge economic or safety consequences if their effect is nottaken into consideration during process or controllerdesign based on a model with uncertainties.

An approach that has much lower computationalexpense than performing exhaustive Monte Carlo simula-tions is to map only the contours of the uncertaintyhyper-ellipsoid obtained for different a levels, from the solu-tion of the worst-case analysis problem (10), based on thepower series expansion representation of the model. Thisapproach is illustrated schematically in Fig. 2 for a case withthree uncertain parameters. With this approach the map-ping of the a levels is performed from the nh dimensionalspace characterized by a chi-squared distribution of nh

degrees of freedom (v2nhðaÞ) to an nw dimensional space in

which the same a-levels are characterized by a chi-squareddistribution with nw degrees of freedom (v2

nwðaÞ), with usu-

ally nw = 1. Hence the probability mapping between thetwo spaces characterized by different degrees of freedomcan be captured by multiplying the obtained worst-casedeviations (dww.c.) with the ratio ðv2

nwðaÞ=v2

nhðaÞÞ1=2. This

method is exact for models that are first-order in the param-eters. For higher-order power series expansions the contour

Fig. 2. Distributional robustness analysis ba

mapping approach is less accurate but much less computa-tionally expensive than the approach described next.

2.2.2. Uncertainty propagation using the Monte Carlo

method with second-order power series expansion

For increased accuracy of the estimated shape of thepdf, the second-order series expansion can also be used ina classical or Latin Hypercube type Monte Carlo simula-tion. In this approach the model is approximated by thesecond-order power series expansion given by (6) wherethe first- and second-order sensitivities are calculated alongthe time-varying control trajectory. The second-order sen-sitivities, M, can be calculated using a finite differenceapproximation from the first-order sensitivities, requiringonly one or two simulations (per parameter) using the non-linear model augmented with first-order sensitivity equa-tions. In the next step, the Monte Carlo method isapplied to the second-order power series expansion withfixed (but time-varying) L and M. This revised MonteCarlo method is orders-of-magnitude computationallymore efficient than applying the brute-force Monte Carlomethod to the original nonlinear model. The computa-tional cost of this second-order approximate Monte Carlomethod is higher than using the analytic pdf for the first-

sed on the contour mapping technique.

Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240 233

order power series approximation but provides a much bet-ter estimate of the shape of the output distribution.

2.3. Uncertainty analysis using polynomial chaosexpansions

An alternative to the power series based techniques is touse polynomial chaos expansions (PCEs). If the parameteruncertainties are described in terms of standard normalrandom variables, the PCE [17] can describe the model out-put w as an expansion of multidimensional Hermite poly-nomial functions of the uncertain parameters h. Forother types of random variables, either different polyno-mial bases (e.g., Laguerre for the gamma distribution,Legendre for the uniform distribution, etc.) or an appropri-ate transformation can be used [18]. Using the Hermitebases in the PCE, the output can be expressed in terms ofthe standard random normal variables {hi} using an expan-sion of order d:

wðdÞ ¼ aðdÞ0 C0|fflffl{zfflffl}constant

þXnh

i1¼1aðdÞi1 C1ðhi1Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

first order terms

þXnh

i1¼1

Xi1

i2¼1aðdÞi1i2C2ðhi1 ; hi2Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

second order terms

þXnh

i1¼1

Xi1

i2¼1

Xi2

i3¼1aðdÞi1i2i3C3ðhi1 ; hi2 ; hi3Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

third order terms

þ � � � ; ð16Þ

where nh is the number of parameters, the aðdÞi1 ; aðdÞi1i2 ; a

ðdÞi1i2i3 ; . . .

are deterministic coefficients in R to be estimated, and themultidimensional Hermite polynomials of degree m ¼ i1; i2;. . . ; inh

, Cmðhi1 ; . . . ; hmÞ are

Cmðhi1 ; . . . ; hmÞ ¼ ð�1Þme1=2hTh ome�1=2hTh

oh1 � � � ohm: ð17Þ

The polynomial chaos terms are random variables, sincethey are functions of the random variables, and terms ofdifferent order are orthogonal to each other (with respectto an inner product defined in Gaussian measures as the ex-pected value of the product of the two random variable,i.e., E½CiCj� ¼ 0 for Ci 5 Cj). In addition, polynomialchaos terms of the same order but with a different argu-ment list are also orthogonal (E½CmðfhgiÞCmðfhgjÞ� ¼0; i 6¼ j). In PCE any form of polynomial could be usedbut the properties of orthogonal polynomials make theuncertainty analysis more efficient. For example, calculat-ing the expected value of both sides of (16) results in the ex-pected value of w being simply E½wðdÞ� ¼ aðdÞ0 C0. Thecalculation of other statistical measures is also significantlysimplified using the properties of orthogonality. Theorthogonal polynomials are derived from the probabilitydistribution of the parameters using the orthogonalitycondition:Z

0

fd:f:ðhÞCiðhÞCjðhÞ ¼ dij where dij ¼ 1 if i ¼ j; dij

¼ 0 if i 6¼ j: ð18Þ

Since C0(h) = 1 the first-order Hermite polynomial canbe calculated from

Rh1

f ðh1ÞC1ðh1Þð1Þ ¼ 0, and the proce-dure can be repeated to obtain all terms in the PCE.The number of coefficients in the PCE depends on thenumber of uncertain parameters and the order of expan-sion (e.g., there are 6 coefficients for two parametersand a second-order PCE and 15 coefficients for afourth-order PCE, whereas for four uncertain parametersthere are 15 coefficients for a second-order PCE and 70for a fourth-order PCE), however, for most engineeringapplications it is not necessary to use higher order thanthree or four. The polynomial chaos expansion is conver-gent in the mean-square sense [18], therefore the coeffi-cients in the PCE are calculated using least-squaresminimization considering sample input/output pairs fromthe model, so that the best fit is achieved between thePCE and the nonlinear model (or experimental data).Because all parameters are considered as standard ran-dom variables, for a more accurate determination of thecoefficients aðdÞi1 ; a

ðdÞi1i2 ; a

ðdÞi1i2i3 ; . . . the probability distribution

of {hi} has to be considered in the selection of the sam-pling points. In principle there are two main methodsfor computing the coefficients of the PCE: (i) the probabi-listic collocation method (PCM) [12,13], and (ii) theregression method with improved sampling (RMIS) [11].Both methods are weighted residual schemes, which differin the way in which the sampling points are chosen. Themethods use the principle of collocation, which imposesthat w is exact at a set of chosen collocation points, thusmaking the residual between the output of PCE and com-plex nonlinear model at those points equal to zero. InPCM, the number of collocation points is set equal tothe number of unknown coefficients, which are found bysolving a set of linear equations generated from the out-puts from the original simulation model. In RMIS, addi-tional collocation points are selected to improve theaccuracy of the computed coefficients. The two methodsalso differ in how the collocation points are chosen. InPCM, the collocation points are selected from the rootsof the orthogonal polynomial of a degree one higher thanthe order of the PCE [19,20]. Which roots are selectedaffects the accuracy of the approximation. In this workthe probability collocation method was used to calculatethe coefficients. An iterative approach to select the orderof the PCE was implemented (see Fig. 3), with automaticderivation of the Hermite polynomials using an algorithmbased on ORTHPOL [21].

Generally, the approaches based on PCE are similar tothe power series expansion-based techniques, since theyalso utilize a simpler representation of the simulationmodel, which can be used to compute the pdf of the out-puts (either directly via the Monte Carlo method or viathe contour mapping approach based on the solution ofthe worst-case optimization). The PCE and PSE can beused to analytically compute statistical measures, suchas the mean, variance, or higher order moments of theoutputs.

Specify uncertainty description for parameters

Derive orthogonal polynomial for the

distribution

Generate 1st order PCE to approximate model

Generate collocation points for model runs

Run model on collocation points and solve for

approximation

Check error of approximation

Is error too large?

Use approximation for uncertainty analysis

Add higher order PCE terms to polynomials

YES

NO

Use contour mapping approach

Monte Carlo simulations

. .1

maxw c

pqdqdy dy

£=

W

Fig. 3. Flowchart of the algorithm used to obtain the PCE approximationof the nonlinear model for distributional robustness analysis.

234 Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240

3. Application to a simulated batch crystallization process

Crystallization from solution is an industrially impor-tant unit operation due to its ability to provide high purityseparation. The control of the crystal size distribution(CSD) can be critically important for efficient downstreamoperations (such as filtration or drying) and product qual-ity (e.g., bioavailability, tablet stability, dissolution rate).Most studies on the optimal control of batch crystallizersfocus on computing the temperature profile that optimizessome property of the CSD [22]. The problem of computingthe optimal temperature profile can be formulated as anonlinear optimization problem, which is then solved usinggeneral-purpose optimization algorithms. The commonlyused approach to describe the temperature trajectory is todiscretize the batch time and consider the temperatures atevery discrete time k as the optimization variables. In thiscase the optimal control problem can be written in the fol-lowing form:

optimizeT ðkÞ

J ð19Þ

subject to :

f ðxÞ ¼

B

Gl0

2Gl1

3Gl2

4Gl3

�qckv3Gl2

Glseed;0

2Glseed;1

3Glseed;2

2666666666666666664

3777777777777777775

ð20Þ

T minðkÞ 6 T ðkÞ 6 T maxðkÞ;

RminðkÞ 6dT ðkÞ

dt6 RmaxðkÞ;

Cfinal 6 Cfinal;max; ð21Þ

where the objective function J is a function of the states,and usually is a representative property of the final CSD.The equality constraints (20) are the model equations, withinitial conditions given in [23], where li is the ith moment(i = 0, . . ., 4) of the total crystal phase (resulted fromgrowth from seed and nucleation), lseed,j is the jth moment(j = 0, . . ., 3) corresponding to the crystals grown fromseed, C is the solute concentration, T is the temperature,r0 is the crystal size at nucleation, kv is the volumetric shapefactor, and qc is the density of the crystal. The rate of crys-tal growth (G) and the nucleation rate (B), respectively, aregiven by [24]:

G ¼ kgSg; ð22ÞB ¼ kbSbl3; ð23Þ

where S = (C � Csat)/Csat is the relative supersaturation,and Csat = Csat(T) is the saturation concentration. Themodel parameter vector consists of the kinetic parametersof growth and nucleation:

hT ¼ ½g; kg; b; kb�; ð24Þ

with nominal values [16]:

hT ¼ ½1:31; expð8:79Þ; 1:84; expð17:38Þ�; ð25Þ

with the uncertainty description of the form (4) character-ized by the covariance matrix [7]:

V�1h ¼

102873 �21960 �7509 1445

�21960 4714 1809 �354

�7509 1809 24225 �5198

1445 �354 �5198 1116

26664

37775: ð26Þ

In the inequality constraints (21), Tmin, Tmax, Rmin, andRmax are the minimum and maximum temperatures andtemperature ramp rates, respectively, during the batch.The first two inequality constraints ensure that the temper-ature profile stays within the operating range of the crystal-lizer. The last inequality constraint ensures that the soluteconcentration at the end of the batch Cfinal is smaller thana certain maximum value Cfinal,max set by the minimumyield required by economic considerations.

Typical crystal size distribution (CSD) parameters ofinterest are the nucleation to seed mass ratio (Jn.s.r.), coef-ficient of variation (Jc.v.), and weight mean size of the crys-tals (Jw.m.s.):

J n:s:r: ¼ ðl3 � lseed;3Þ=lseed;3 ð27ÞJ c:v: ¼ ðl2l0=ðl1Þ

2 � 1Þ1=2 ð28ÞJ w:m:s: ¼ l4=l3 ð29Þ

Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240 235

The optimal temperature trajectory in Fig. 4 used tocompare the uncertainty analysis methods was computedby solving the optimal control problem (19)–(21) forJ = Jn.s.r. with the nominal parameter h, which led to adecrease of more than 20% in the nucleation to seed massratio compared to the linear cooling profile. The distribu-tional uncertainty analysis approaches based on the powerseries approximation and polynomial chaos expansion,respectively, were evaluated in comparison to Monte Carlo(MC) simulations based on the full nonlinear model.80,000 random parameter sets with mean hi and covarianceVh were generated from the multivariate distribution usingthe Cholesky decomposition of the covariance matrix.With the random parameter vectors, Monte Carlo simula-tion was performed using the full dynamic model. Then,the frequency that the simulation output from the MonteCarlo simulation fell in the confidence interval obtainedfor a certain a level with the power series approach wascomputed for all states and outputs. The results obtainedwith the first-order analysis approach for the output vari-ables are shown in Fig. 5. The accuracy of the first-orderapproach is very good with a slightly decreasing tendencyfor large a values. This can be explained by the cumulativeeffect of the truncation error due to the first-order

0 40 80 120 16028

29

30

31

32

Time (min)

T (

°C)

Fig. 4. Linear cooling and optimal temperature profile for J = Jn.s.r..

0 20 40 60 80 1000

20

40

60

80

100

α [%]

Fre

quen

cy fr

om M

C [%

]

Jc.v.

Jn.s.r.

Jw.m.s.

Fig. 5. Comparison between first-order power series analysis and theMonte Carlo method applied to the full nonlinear model. The dashed linecorresponds to the ideal case.

approach (which increases when a increases) and the taileffect of the Monte Carlo simulation (as seen on Fig. 1).Although there is a slight decrease in the accuracy of thefirst-order approach for large a, in the case of certain pro-cess outputs the first-order expansion was quite accuratefor estimating the size of confidence intervals on the CSDobjectives. As seen next, the first-order expansion is notaccurate for estimating the shape of the pdfs on the CSDobjectives.

For a better comparison of the uncertainty analysismethods, the entire pdfs of the states and outputs are com-puted based on the frequency histograms for all modelstates and outputs at the end of the batch. The histogramswere obtained by splitting the range of the values in equal-sized bins (called classes), and then counting the number ofpoints that fall into each bin. The probability distributionfunctions were calculated from the frequency histogramsof the simulations, as the count in the class divided bythe number of observations times the class width. For thisnormalization, the area (or integral) under the histogram isequal to one, and the resulted normalized histogram resem-bles the real pdf. Fig. 6 shows the histograms obtained viathe Monte Carlo method applied to the full nonlinearmodel, using 80,000 parameter sets. The resulted distribu-tions are not Gaussian, showing the effect of the nonlinear-ity on the uncertainty propagation. Some of the states arecloser to the form of the normal distribution, e.g., l3,whereas others, e.g., l0, show large deviations from aGaussian distribution. Although the Monte Carlo methodprovides a good estimate for the shape of the nonlinear dis-tributions, it requires a large number of sampling pointsand very high computational cost (see Table 1), whichmakes it inappropriate for on-line applications such asreal-time robust feedback control and inconvenient foroff-line applications such as robust batch recipe design.

The first-order PSE approach estimates the output dis-tributions (Fig. 7) at very low computational cost (Table1). Since the first-order PSE approach uses a linear approx-imation of the nonlinear model, the output distributionsare Gaussian, but still provide a good approximation ofthe real distributions for most of the states and outputs.This approximation is accurate enough, for example, forrobust controller design, or minimum variance control,where the objective is to design a controller that decreasesthe variations around the nominal performance index. It isan acceptable approximation that, if the variance of theGaussian approximation of the nonlinear distribution isdecreased, a similar effect will be achieved for the realdistribution.

There certainly exist some finite-time processes in whichfirst-order analysis will not be accurate enough. Also, accu-rate quantification of the risk of producing batches withpoor product characteristics and accurate assessment ofthe performance of fault diagnostic systems requires amore accurate quantification of the output pdfs. In thiscase the higher order power series expansion or polynomialchaos expansion give more accuracy while keeping the

Fig. 6. Probability distribution functions at the end of the batch obtained via the Monte Carlo method applied to the full nonlinear model.

Table 1Computational cost of the various approaches to compute the distributionfor all process outputs during the entire batch (on a P4-1.4 GHzcomputer)

Method Computationaltime

Monte Carlo simulation with full dynamic model(80,000 points)

8 h

First-order approach (50 a levels) 1 sMonte Carlo applied to second-order series expansion

model (80,000 points)4 min

Second-order polynomial chaos expansion 20 s

236 Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240

computational cost low. Applying the Monte Carlomethod to the second-order PSE results in a more accurateapproximation of the nonGaussian distribution (Fig. 8)with relatively low computational cost (Table 1). In oursimulations the sensitivity jacobian L was calculated byintegrating the sensitivity Eq. (9) which were 32 (nx · nh)ODEs. The computational cost was reduced by computingthe matrices Jx = df/dx and Jh = df/dh analytically (con-sidering r0 = 0):

A¼

0 0 0 B=l3 0 bB=ðC�CsÞ 0 0 0

G 0 0 0 0 gGl0=ðC�CsÞ 0 0 0

0 2G 0 0 0 2gGl1=ðC�CsÞ 0 0 0

0 0 3G 0 0 3gGl2=ðC�CsÞ 0 0 0

0 0 0 4G 0 4gGl3=ðC�CsÞ 0 0 0

0 0 �3qckvG 0 0 �3qckvgGl2=ðC�CsÞ 0 0 0

0 0 0 0 0 gGlseed;0=ðC�CsÞ 0 0 0

0 0 0 0 0 2gGlseed;1=ðC�CsÞ 2G 0 0

0 0 0 0 0 3gGlseed;2=ðC�CsÞ 0 3G 0

266666666666666664

377777777777777775

ð30Þ

Jh ¼

0 0 B lnðSÞ B=kb

l0G lnðSÞ l0G=kg 0 0

2l1G lnðSÞ 2l1G=kg 0 0

3l2G lnðSÞ 3l2G=kg 0 0

4l3G lnðSÞ 4l3G=kg 0 0

�3qckvl2G lnðSÞ �3qckvl2G=kg 0 0

lseed;0G lnðSÞ lseed;0G=kg 0 0

2lseed;1G lnðSÞ 2lseed;1G=kg 0 0

3lseed;2G lnðSÞ 3lseed;2G=kg 0 0

266666666666666664

377777777777777775

:

ð31Þ

To obtain the matrix of second-order sensitivities, M, for-ward finite differences was used, which requires only oneadditional simulation with the augmented model for eachparameter. For the Monte Carlo simulations for a fixedcontrol trajectory, L and M are computed off-line andMonte Carlo simulations are performed using the PSE,which is computationally much more efficient than usingthe full integration of the nonlinear model expressed bythe set of ODEs.

As with the PSE, the PCE can be used with the MonteCarlo method, or contour mapping which solves of theoptimization

dww:c:ðaÞ ¼ maxkWhdhk61

dwPCE; ð32Þ

for different values of a, where dwPCE is calculated for thePCE. The order of the PCE was determined to be two usingthe iterative approach in Fig. 3 (including third-order termsdid not increase the accuracy of the approximation signif-icantly). Since there are four uncertain parameters, the sec-

0 2000 4000 60000

0.5

1

x 10-3

μ0

0 2 4 60

0.5

1

μ1×10-54 6 8 10

0

0.2

0.4

0.6

0.8

μ2×10-7

2.95 3 3.050

20

40

μ3×10-101 1.2 1.4 1.6 1.8 2

0

2

4

μ4×10-130.429 0.4295 0.43 0.4305 0.4310

1000

2000

C

2000 2100 2200 2300 24000

0.005

0.01

μseed,1

2 2.2 2.4 2.6 2.8 30

2

4

6

μseed,2×10-62 2.5 3 3.5

0

1

2

3

μseed,3×10-9

0.8 0.9 1 1.1 1.20

5

10

15

Coeff. var.6 8 10 12

0

0.5

1

Mass ratio400 500 600 7000

0.005

0.01

0.015

Weight mean size (μm)

Fig. 7. Probability distribution functions at the end of the batch obtained from the first-order power series expansion (PSE).

Fig. 8. Probability distribution functions at the end of the batch obtained via the Monte Carlo method applied to the second-order PSE.

Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240 237

ond-order PCE required the determination of 15 coeffi-cients (according to N ð2Þc ¼ 1þ 2nh þ nhðnh � 1Þ=2). Thesecond-order four-dimensional PCE is

wð2ÞðhÞ ¼X14

i¼0

aiCi; ð33Þ

where the Hermite polynomials are in Table 2. The coeffi-cients ai were computed using the probabilistic collocationmethod with the collocation points obtained from the roots

of the third-order Hermite polynomials [19,20]. The opti-mization problem (32) was convex for the four-dimensionalsecond-order polynomial as the objective function and sowas easy to solve (see Fig. 9). The second-order PCE veryaccurately captured the shape of the pdfs at the end of thebatch (compare Figs. 6 and 9), resulting in a more accurateestimation of the effects of parameter uncertainties than thesecond-order PSE at a lower computational cost.

The pdfs of the moment l0, which shows the strongestdeviation from the Gaussian distribution, are plotted for

Table 2The Hermite polynomials for the second-order four-dimensional PCE

ith Polynomial chaos Ci Order of polynomial chaos

0 1 01 h1 12 h2 13 h3 14 h4 15 h2

1 � 1 26 h1h2 27 h1h3 28 h1h4 29 h2

2 � 1 210 h2h3 211 h2h4 212 h2

3 � 1 213 h3h4 214 h2

4 � 1 2

238 Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240

the first-order PSE and second-order PCE in Fig. 10 withthe pdf from the Monte Carlo method applied to the fullnonlinear model. As noted before, while the first-orderPSE gave acceptable accuracy (which is even better forthe rest of the states and outputs) for most purposes, amore accurate estimate of the tails of the distribution canhave significant economic or safety implications, especiallyin the case of robust design. If the performance specifica-tion was in terms of l0, Fig. 10 illustrates that, if therewas a high penalty on exceeding the specification limits,then a design based on the first-order PSE would leadto a more conservative result with respect to the lowerspecification limit, whereas it underpredicts the probabil-ity of events with large values of l0 which could havesafety implications in particular applications, such aspharmaceutical crystallization. The second-order PCE

Fig. 9. Probability distribution functions at the end of the batch obtained viaexpansion.

based approach captures very well the shape and tails ofthe distribution.

Unlike sensitivity analysis, these uncertainty analysisapproaches provide the variation of the whole distributionfor all states and outputs along the entire batch. This isillustrated in Fig. 11, which shows the pdfs for the perfor-mance index Jw.m.s. with the various uncertainty analysismethods. As noted before, the power series and PCErobustness analysis methods have low computational costs,therefore they can be used easily for the assessment andefficient design of robust control systems. The simplifieddependency of these expansions on the parameters leadto simpler optimization of the parameters than obtainedby direct robust controller synthesis, which usually leadsto nonconvex problem formulations. Robust controllerdesign can be formulated as a classical minmax optimiza-tion or as a multiobjective optimization problem whereone objective accounts for the nominal performance andthe other (usually a measure of the variance of the distribu-tion) accounts for robustness [25,26]. Both uncertaintyanalysis methods can be used to efficiently calculate therobustness term. Additionally, since the PCE approachprovides a means for accurate estimation of the shape ofdistribution, it can form the basis of a procedure fordesigning controllers to shape the distribution, whichwould provide more flexibility in addressing uncertaintythan worst-case or minimum variance control. This is espe-cially motivated by the asymmetry in the economic costfunctionals of processes operated near safety, environmen-tal, and quality constraints; for example, designing a con-troller to shape the distribution in relation to the costfunctional rather than merely reducing variance wouldmore closely map controller design to economic objectives.

the Monte Carlo method applied to the second-order polynomial chaos

0 500 1000 1500 2000 2500 3000 35000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10-3

μ0

pdf

Opt

imal

spe

cific

atio

n

2nd

ord.

PC

E

1st o

rd. P

SE

Profit

Opt

imal

spe

cific

atio

n

Spe

c. w

ith 2

nd o

rd. P

CE

Spe

c. w

ith 1

st o

rd. P

SE MC with NL model

1st order PSE2nd order PCE

Fig. 10. Potential effect of better distribution shape estimation on theprofit/safety in the design under uncertainties. Comparison of theperformance between the Monte Carlo (MC) method applied to the fullnonlinear (NL) model, and the first-order power series expansion (PSE)and second-order polynomial chaos expansion (PCE) methods.

050

100150

200300

400500

600700

0

0.01

0.02

0.03

0.04

time (min)weight mean size (μm)

pdf

Fig. 11. Variation of the distribution of the weight mean size along thebatch, computed using various uncertainty analysis methods. The solidline is for the Monte Carlo method applied to the full nonlinear modelwith 80,000 sets of parameter, the dots are for the first-order power seriesmethod, the circles are for the Monte Carlo method applied to the second-order power series expansion, and the x-marks are for the polynomialchaos expansion method.

Z.K. Nagy, R.D. Braatz / Journal of Process Control 17 (2007) 229–240 239

4. Conclusions

This paper provides an overview of several computa-tionally-efficient robustness analysis approaches, focusingprimarily on computing probability distributions on statesand outputs. The approaches are based on the approximaterepresentation of the full process model using power seriesor polynomial chaos expansions, and provide a qualitativeand quantitative estimation of the effect of parameteruncertainties on the states and output variables along abatch. The computational cost of the robustness analysismethods are significantly lower than the classical Monte

Carlo method applied to the full nonlinear model. Applica-tion of the algorithms to a simulated batch cooling crystal-lization process provided the recommendations: (1) forsome systems, the first-order series analysis is accurateenough for the design of real-time robust feedback control-lers, but the accuracy of the first-order expansion should beconfirmed a posteriori using Monte Carlo simulation or ahigher order PCE method, and (2) even with a relativelylow-order approximation, the polynomial chaos expan-sions can provide a very good approximation of the shapeand tails of the output and states distribution for batchprocesses, providing a generally applicable approach foruncertainty propagation in robust batch control anddesign.

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