Linking Models and Experiments - Industrial & Engineering

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Linking Models and Experiments Dominique Bonvin, Christos Georgakis, Constantinos C. Pantelides, Massimiliano Barolo, Diogo Rodrigues, René Schneider, Denis Dochain, and Martha A. Grover Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b04801 • Publication Date (Web): 25 May 2016 Downloaded from http://pubs.acs.org on May 26, 2016

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Industrial & Engineering Chemistry Research

Linking Models and Experiments D. Bonvin a,b, C. Georgakis *,###, C. C. Pantelides c, §, M. Barolo d, §, M. A. Grover e, §, D. Rodrigues a, §, R. Schneider f, § and D. Dochain g, § a

Ecole Polytechnique Fédérale de Lausanne, Switzerland b Tufts University, Medford, MA, USA c Imperial College London, UK and Process Systems Enterprise Ltd, UK d University of Padova, Italy e Georgia Institute of Technology, Atlanta, GA, USA f RWTH Aachen University, Germany g Université Catholique de Louvain, Belgium b

Session organizer, #Speaker & Conference Organizer, § Participant

Abstract This position paper gives an overview of the discussion that took place at FIPSE 2 at Aldemar Resort, east of Heraklion, Crete, in June 21-23, 2014. This is the second conference in the series “Future Innovation in Process Systems Engineering” (http://fi-inpse.org), which takes place every other year in Greece, with the objective to discuss open research challenges in three topics in Process Systems Engineering. One of the topics of FIPSE 2 was the issue of “Linking Models and Experiments”, which is described in this publication. Process models have been used extensively in academia and industry for several decades. Yet, this paper argues that there are still substantial challenges to be addressed along the lines of model structure selection, identifiability, experiment design, nonlinear parameter estimation, model validation, model improvement, on-line model adaptation, model portability, modeling of complex systems, numerical methods, software environments and implementation aspects. Although there has been an exponential increase in the number of publications dealing with “modeling”, the majority of these publications do not use sound statistical tools to evaluate the model quality and accuracy and also present modeling as a non-iterative task. As a result, the models often have either too few or too many parameters, thus requiring trimming down or enhancing before they can be used appropriately. Also, this position paper argues that the models should be developed with a purpose in mind, as, for example, different models are needed for design, control, monitoring, and optimization.

*Correspondence should be sent to: [email protected]

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1. Introduction The rapid advances of computational methods make the use of mathematical models increasingly pertinent and useful. Figure 1 illustrates that there has been an exponential growth in the number of publications with “modeling” in the title. There are, in principle, two sources of information for building process models, namely, (i) the knowledge of experts in the form of laws of nature that have been worked out by generations of scientists, and (ii) the system itself from which data can be collected 1. We will consider here three classes of process models that are labeled knowledge-driven, data-driven and hybrid (or grey-box). Knowledge-driven models rely on our understanding of the inner working of the process; these models have also been called “first-principles models”. For chemical engineering processes, they typically consist of momentum, material and energy balances that include constitutive relationships to model kinetic, transport and thermodynamic phenomena 2-4. Experimental data are then used to estimate the values of the model parameters that are not well known a priori 5-6. In contrast, data-driven models largely ignore physical knowledge and construct mathematically the input-output relationships that are observed experimentally. Once a mathematical structure is chosen, these models depend exclusively on the data at hand 7,8. Hybrid models are a combination of the first two classes, which implies that certain model components represent the knowledge we have about the inner working of the process, while the remaining components are determined from process measurements 9-10. Note that each model class contains different model types, such as static or dynamic, deterministic or stochastic, continuous-time or discrete-time, lumped or distributed, change or discrete-event driven, state-space or input-output, linear or nonlinear, in the time domain or in the frequency domain. Regardless of its class and type, a mathematical model consists of a set of equations that define its structure and a set of corresponding parameters. At this point, the reader is invited to notice the distinction that is made in this paper between the concepts of model class, model type and model structure. Mathematical models are central to process design, process analysis and process operations 11-12. Certain models are kept relatively simple and describe only the main features of the process, often only at steady state. Other models are quite elaborate and cover a wide range of phenomena in a socalled multi-scale approach 13-15. It is often stated “Modeling is an Art”. Indeed, in addition to the extensive computational tools that are used, modeling involves several synthetic and conceptual tasks, such as the choice of the model type and structure and the appropriate degree of complexity. Another key issue is “Modeling for a Goal”, which emphasizes the fact that the model is a means to an end and not the final object of investigation. Hence, for the same process, different models might be necessary for different objectives, such as planning, scheduling, simulation, control and optimization. Furthermore, the model is not an entity that is determined once for all, but rather it should be allowed to evolve when, for example, the process varies with time or is operated differently. This is the field of “Model Maintenance”. This position paper addresses important issues in process modeling. In particular, it presents the model classes, the modeling methodology and a list of open research problems. We hope that this paper will be a source of inspiration for people in both academia and industry, and that it will help shape research efforts in the field of process modeling within the important area of Process Systems Engineering 16. Page 2 of 27


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2. Model Classes This section briefly describes the three model classes and their relative merits.

2.1. Knowledge-driven Models Knowledge-driven models are the most desirable of all models because they provide a systematic way of organizing our physical understanding of the process. For example, if we do not understand all the constituent parts, the development of a knowledge-driven model will force us to invest time and effort to understand and describe the missing parts. Knowledge-driven models often represent continuous processes that are used to manufacture large quantities of a given product. This is important, as the effort needed to develop such models is often recovered by even small operational improvement. It turns out that knowledge-driven models, which can be static or dynamic, are most often nonlinear, continuous-time, deterministic state-space representations 17. The parameters of knowledge-driven models often have a physical meaning, which might facilitate their estimation, for example allowing one to reject values that are unreasonable from a physical point of view.

2.2. Data-driven Models This class of models is very useful when the inner workings of a process are not well understood. They require the availability of the physical process under investigation so that experiments can be performed, data collected, and the model identified. Data-driven models are therefore nothing else than a convenient representation of these data. Data-driven models include, among others, (i) the Response Surface Methodology (RSM) models related to the design-of-experiments methodology 18, (ii) the transfer-function (TF) models that are often used in the design and tuning of process controllers 11 , and (iii) principal component analysis (PCA) and partial least-squares (PLS) regression models that are based on multivariate analysis and can be used to implement process monitoring 19, soft sensing 20, and statistical process control 21. Most data-driven models are linear in the parameters, although the input-output behavior might be quite nonlinear 18, 22. Often, the parameters of data-driven models do not have a clear physical meaning. Notable exceptions are transfer-function model parameters, such as gains and time constants, and latent-variable model parameters (e.g., loadings), which can be analyzed to uncover physical relationships hidden within the data.

2.3. Hybrid Models A hybrid (or grey-box) model is one that combines partial physical knowledge with data-driven parts that are constructed using experimental data. The distinction with knowledge-driven models is subtle but important: in a knowledge-driven model, all the mathematical relationships are known a priori, usually being based on existing physical and engineering knowledge, thus leaving only the numerical values of some of the model parameters to be identified from experimental data. In contrast, in a hybrid model, some of the mathematical relationships are unknown and therefore need to be identified as a complementary data-driven part to the incomplete knowledge-driven model. As an example, let us consider a reactor in which a set of reactions takes place and about which we know only part of the reaction pathways together with the corresponding rate laws. In such a case, material balances can be written for the known part of the reaction network. The unknown part of the stoichiometry and the corresponding kinetics might be more expediently represented through a data-

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driven model component. If sufficient experimental data are available to construct the data-driven model complement, then the overall model can be classified as a hybrid one. This situation is very common in the modeling of complex pharmaceutical reaction networks, where the reactions leading to several unwanted impurities are not as well understood as the main reactions 23. Often, such datadriven complementary parts will be in the form of an RSM model.

2.4. Relative Merits of the three Model Classes Knowledge-driven models are the preferred ones since they typically allow extrapolation beyond the range of the data currently available for the system under investigation. Knowledge-driven models are possible when the process is well understood and can be represented mathematically. The development of such models requires a substantial investment in time and effort. These models are rarely fully predictive a priori, as some of the model parameters need to be estimated from experimental data. In comparison, data-driven models require only minimal understanding of the inner working of the process, which makes their derivation easier. They are interpolative in nature. Their development requires substantially less effort. Hybrid models are somewhere in-between, as they require less process knowledge than knowledge-driven models but more than data-driven ones. To our surprise, hybrid models have not been pursued as systematically as the other two classes, despite the fact that they can take advantage of some process knowledge, which is left unutilized when datadriven models are used.

3. Modeling Methodology This section first discusses the necessity to have a comprehensive modeling framework that can guarantee that the developed models are statistically sound. Then, we describe the model building steps for both the knowledge-driven and data-driven approaches and elaborate on how such a modeling framework can be applied. The construction of hybrid models combines the two approaches, with initially the building of a knowledge-driven part and then the use of (residual) data to generate the data-driven part.

3.1. Modeling Framework Models are tools toward a goal. Hence, it is important to (i) clearly define the purpose of the model, and (ii) properly select the class, type and structure of the model to be developed. Another important issue is whether the model is developed to work in interpolation (within the calibration set) or in extrapolation (possibly also outside the calibration set). The modeling framework includes the following steps 2: 1. Model structure selection. The reality is often quite complex, and a fair deal of abstraction is necessary to be able to represent the dominant phenomena while discarding those that can be neglected. Working hypotheses are introduced and the structure of the mathematical model is proposed based on the modeler’s preferred trade-off between accuracy and complexity 2, 17. It is also often necessary to discriminate between rival structures that could potentially describe the same system. 2. Experimental data and parameter estimation. Even if the model is fully knowledge-driven, it will typically be necessary to estimate some of the model parameters from measurements.

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Data can be collected in a number of different ways, using for example the Design-ofExperiments (DoE) methodology 18, 22. A set of historical process data can also be used, provided it is sufficiently informative. There are many parameter estimation methods available, of which the least-squares approach is the most popular. Other approaches include the leastabsolute-value approach, maximum-likelihood estimation, Bayesian estimation and various types of robust estimation 6. 3. Model validation and model improvement. Model validation and cross validation are important issues that, if performed appropriately, bring substantial credibility to the developed model 24-25. Strictly speaking, one can never prove conclusively that a model is valid on the basis of existing data, since additional data might well invalidate the model. Instead, one determines whether or not, with a given probability, the proposed model can be invalidated by the existing data. Furthermore, in order to test the predictive (extrapolative) capability of the model, a certain fraction of the experimental data should not be used for parameter estimation but reserved for the “model validation” step. The modeling task is often considered complete at this stage, which is unfortunate because the following important questions remain unanswered: • Question A - SIGNIFICANCE: Are all the estimated parameters significant? o The estimated values of the model parameters should always be accompanied with some measure of their uncertainty, possibly in the form of a confidence region, with a chosen level of certainty, say 95%. If the confidence interval for a given parameter is large despite appropriate experimental design, it is then best to fix that parameter at some value in the interval and not include it in the set of estimated parameters. In particular, if the confidence region incudes zero, it is appropriate to set the parameter value to zero and discard the corresponding term from the model. When some parameters are found of limited significance, it is justified to search for a parameter vector of lower dimension providing nearly the same validity. • Question B - UNIQUENESS: Is it possible that significantly different sets of parameter values provide nearly the same residual sum of squares? o This happens when the confidence region of the parameter set is large, often an indication of either poor experimental design or the presence of too many parameters (which leads to highly correlated parameters). In the latter case, one could try to simplify the model by reducing the number of parameters. How this is achieved, will be detailed below. • Question C - SUFFICIENCY: Is the residual sum of squares comparable in size to the sum of squares related to the normal variability of the process? o If the residual sum of squares is statistically larger than the normal variability of the process, then the model has failed to interpret all the non-random variability in the data. In this case, the model needs to be enriched with additional components and the enlarged set of parameters estimated afresh. Model validation also involves evaluating a posteriori, on the basis of model prediction, the appropriateness of certain simplifying assumptions such as lumping of certain effects, isothermal behavior, or negligible heat loss. If these assumptions were used for model

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development, are they justified? And if they were not used, should they be introduced? Also, the residuals can be utilized to assess the quality of the identified model. For example, the residuals can be correlated with specific elements in the model to assess whether these elements require a more detailed description. The residuals can also be plotted as functions of time, or against the fitted response or the regressor variables 6 . Finally, this type of a posteriori analysis is quite useful for conducting another set of experiments aimed at improving model quality, by helping decide where to perform additional experiments. The modeling framework is described next for the cases of both knowledge-driven and data-driven models.

3.2. On Building Knowledge-driven Models The construction of knowledge-driven models starts by analyzing the dominant phenomena at play, making simplifying assumptions and writing material, momentum and/or energy balances around each part of the system under investigation. For example, these parts can be single units such as reactors, heat exchangers and separators in a processing plant, or subsystems in a physiological system. Key questions that a modeler needs to address include (i) what is known about the system that can be described mathematically, and (ii) which are the most appropriate components to include in the model. In principle, these questions should be examined explicitly up front. In practice however, there is often not enough information to make the proper choices. If there are too few model components, the model will have difficulty representing the observed data. If, on the other hand, the model has more than the required complexity, it might be very difficult to estimate the values of all the parameters unambiguously. Einstein once said “Everything should be made as simple as possible, but not simpler” 26. The minimal complexity of a model cannot be determined a priori, as it is the result of an iterative process that starts from a model candidate. One then modifies the structure of the initial model by enriching it with additional components or by trimming it down to a simpler form. Statistical tools are available that can indicate the need to either enrich the model with additional components or simplify it 25, 27. This is also the field of model discrimination, for which model-based design of experiments can be used 28. If the model structure is not easily discoverable in a single step, how does one evolve from the initial structure to an improved one? This involves systematic analysis of the estimated model, for which the modeler needs to answer the aforementioned three questions: 1. Examining Question A might result in dropping one or several components from the initial model because the corresponding parameters are found insignificant. 2. Answering Question B might lead to the conclusion that two or more model parameters, which were thought to be independent from each other, are highly correlated. This conclusion does not eliminate a model component but reduces the number of parameters to be identified. 3. Addressing Question C brings forth the possibility that the proposed model is not sufficiently detailed to fully represent the non-random variability observed in the data. In such a case, additional components need to be added to the model, which represents a synthetic task that uses the modeler’s intuition and understanding of the process. Even if the three questions are dealt with correctly, the uncertainty in the estimated model parameters might still be unacceptably large, thereby causing the model predictions to have a large confidence

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region. In such a case, additional experiments are needed to improve the model. These experiments should be designed so as to reduce the uncertainty of model parameters or increase the predictive capability of the model in the region of interest 18, 22. Once additional data are collected and processed, Questions A, B and C should, of course, be re-examined. In this context, it is important to keep in mind that (i) the modeling procedure is intrinsically iterative, and (ii) a model can never be validated as “true” through comparison with data. Instead of saying that one aims at discovering the true model, it is more accurate to say that one aims for a model that will not be proven wrong or invalid by the available data.

3.3. On Building Data-driven Models The model-building step in data-driven models is intimately linked to the analysis of the measured data. Several types of data-driven models exist, and they are very useful in many aspects of Process Systems Engineering. For example, TF models are commonly used in process control. PCA 29 and PLS 30 regression models and neural networks 31 are used to construct linear and nonlinear relationships between process inputs and outputs; these models almost exclusively describe the steady-state behavior, as they do not involve time explicitly. RSM models result from the application of the DoE methodology 18, 22. This methodology, as well as its extension to handle dynamic experiments (DoDE) 32-34 , include three components, namely, the carefully chosen experimental runs, the interpolative RSM model and the analysis of variance. The latter uses statistical tests to determine whether each parameter in the model is significant and whether the resulting model sufficiently explains the collected data. This way, the structure of the RSM model can be inferred iteratively. Because the RSM models are linear in the parameters, these three tasks are much easier than the corresponding ones for knowledge-driven models, whose parameter dependence is often nonlinear. The Analysis of Variance (ANOVA) plays a central role in assessing the quality of a model before a modeling task can be declared complete 22. In particular, ANOVA helps address Questions A, B and C on significance, uniqueness and sufficiency. The use of ANOVA is described next in the context of datadriven modeling using a least-squares approach. At the same time, we would like to motivate its use also in the context of knowledge-driven models. 1. Let us assume that we have made n experiments in which we have varied the values of the input variables (or factors) in some systematic manner. By proper transformation, the input variables are made dimensionless so that all their input values are in the [-1, +1] range. These dimensionless variables are called coded variables. We also assume that a few experiments, usually 3 or 5, are replicated at the same input conditions. Often, these replicated experiments are performed at the center point of the experimental region, where the coded variables are equal to zero. These replicate experiments are essential in estimating the magnitude of the normal variability of the process via the sum of squares corresponding to random errors, SSE . 2. We first postulate a minimal model with no input, labeled M0, compute the global mean of all the outputs and calculate the related sum of squares involving the n measurements and the global mean, SS M 0 . We then perform a statistical F-test to determine, with a certain level of confidence, whether the two sum of squares SS E and SS M 0 are statistically different 22. If they are, Model M0 is insufficient to represent the measured data, and a model relating the inputs to

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the output needs to be developed. Such models are typically linear in parameters and either linear or quadratic in inputs. The parameter values are calculated as the solution to a linear least-squares problem. A series of statistical t-tests helps determine the significance of each parameter estimate, with the insignificant ones being removed from the model. This is commonly done via a step-wise regression algorithm 35. This task answers Question A on significance. 3. With only the significant parameters retained in Model M, the least-squares algorithm recomputes the parameter values and the unmodeled sum of squares, SSUM , which represents the variability in the data that the model does not explain. With model M at hand, one can determine whether the set of optimal parameter values is unique by computing the Hessian of

SSUM with respect to the model parameters and evaluating its eigenvalues. If the Hessian is ill conditioned, two or more parameters are highly correlated and should not be considered as independent. Information on these parameters is provided by the corresponding eigenvectors. If some of the parameters are highly correlated, the set of parameters should be reduced accordingly. This test can be done for both data-driven and knowledge-driven models, in the latter case using a local linearized approach around the least-squares solution. This answers Question B on uniqueness. 4. At this stage, the model includes only significant and fairly uncorrelated parameters, but the modeling task is not complete. We still need to determine whether the unmodeled sum of squares SSUM is of the same order of magnitude as the normal variability of the process, SSE . This is done by a goodness-of-fit F-test and provides an answer to Question C on sufficiency. If the test fails because the unmodeled sum of squares SSUM is larger that the normal variability expressed by SS E , this calls for a “richer” or enhanced model. In data-driven models this enrichment is achieved by inserting additional terms of polynomial character in the inputs. Unfortunately, this task is largely ignored in most published accounts of knowledge-driven models. How this synthetic task should be performed in knowledge-driven models is an open research issue. If it is ignored, then the uncertainty in model prediction will be significantly larger than it needs to be!

4. Open Research Problems The open problems are arranged along the lines of the modeling steps discussed in Section 3.1; in addition, there is a discussion of implementation aspects. Note that the order of the problems and issues listed next is not necessarily the order in which they need to be tackled in practice. Since the modeling procedure is iterative, these issues might appear several times as part of the same modeling task.

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4.1. Modeling Task and Model Structure Modeling for a Goal Parameter identification is often done via least-squares minimization of residuals, which means that the outputs predicted by the model need to match the measured outputs. Obviously, this is a nice scheme when the objective of the model is to predict the output behavior. Unfortunately, this is no longer true when the model is used for other purposes, such as control and optimization. For example, a good model for control is a model that is able to predict the sensitivity of the controlled process at its crossover frequency, which is critical in avoiding closed-loop instability. If the model parameters have been identified with excitation in other frequency ranges, such as low frequencies for a static model, the resulting model might be of little value for control purposes! This idea has been studied extensively in the 90's in the area of system identification and control under the label ”identification for control” 36-37. Similarly, a good model for optimization is a model that is able to predict the optimality conditions for the optimization problem at hand, namely, the Karush–Kuhn–Tucker conditions for a static optimization problem 38, and the Pontryagin's Minimum Principle conditions for a dynamic optimization problem 39. Since the model is only a means to an end, the mean must be tailored to the end. In the context of “modeling for optimization”, the synergy between the modeling and optimization steps is improved by reconciling the objective functions of the two problems 40. In other words, what appears to make a lot of sense is often overlooked, mostly because the quantities that are readily available are the measured outputs and not the closed-loop sensitivity in a certain frequency range or the optimality conditions that consist of active constraints and reduced gradients. Hence, there seems to be a lot of potential for linking the two tasks of model identification and model usage more closely together. Along the same line, one can expect greater benefits when a good model is used in a simple control scheme than when a poorer model is embedded in an advanced control scheme. Hence, in addition to developing sophisticated control and optimization schemes that require the availability of accurate full-blown models for state estimation and input computation, research should focus on systematic approaches to obtain more accurate specific models for a given objective. Note also that the concept of “modeling for a goal” does not necessarily call for a different model structure. The change is very subtle and often simply calls for a more “goal-oriented” design of experiments or the use of more “goal-oriented” measurements. Model Reduction Model reduction broadly refers to the task of reducing the number of terms in a model, while not sacrificing significant accuracy in prediction. In many ways this is conceptually related to Questions A and B, such that a separate model-reduction step is not necessary, once insignificant and correlated parameters have been taken into account. However, a model may have been constructed for a wide range of behaviors, but is later needed for a more limited set of predictions. In particular, model reduction has historically been framed from an input-output perspective for the purpose of feedback control. A dynamic model might have originally been constructed to represent many manipulated and

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controlled variables, more than are needed for a specific control task. A model reduction step may then be applied to this more expansive model in order to generate a simpler model form, for the purpose of capturing a smaller subset of input-output relationships with a minimal mathematical form. Rigorous proofs and computational tools exist for model reduction of linear dynamic systems 41, but chemical processes are often highly nonlinear, in which case a general scalable approach remains elusive 42-44. Model reduction may be broken down into two steps, namely, state reduction and dynamic model construction. Note that subsequent validation is also necessary, as for any model. To accurately predict a dynamic input-output relationship, it may or may not be necessary to include all of the internal dynamic variables from the original “full” model. Depending on the coupling of the internal dynamic variables and their relationships to the measured outputs of interest, some of the internal dynamic variables may be eliminated or combined. The most common method for state reduction is the use of principal component analysis to identify a lower-dimensional linear subspace and coordinate representation. Nonlinear methods such as neural nets and IsoMap have also been proposed, but do not scale as well to large systems 45. In either case, the determination of which modes to keep is usually based on variance arguments, with no consideration of the dynamic relationships in the data. In practice, it is often not clear how many modes to keep, or even if this variance metric is particularly relevant to predicting dynamic behavior (i.e. small perturbations could later grow into large features). In addition, the variance criterion can be manipulated by scaling and weighting of the measured values. In practice, this human intervention is used for tuning and even to insert domain knowledge about what is “really important” for a particular system. A second step in model reduction is the mathematical representation of the dynamics in this reduced set of state coordinates. For linear systems, the mathematical structure is still linear, but for nonlinear systems the derivation of a reduced mathematical structure may be unwieldy or intractable. As a result, empirical model structures may be a more attractive approach, ranging from global polynomial structures to locally based Gaussian process models (GPM). An advantage of a local model is that it can better represent the observed data (coming from simulations of the full model or from experiments). These models are often referred to as meta-models or surrogate models, especially within the optimization community 46. The GPM also provides prediction variance, although GPM does not scale well to very large datasets. Markov state models provide another data-driven and local approach to dynamic modeling, which also capture uncertainty (due to stochastic effects) through a discrete probability distribution. Thus, even with a knowledge-driven “full” model, a reduced model may either be obtained analytically, or else through a data-driven approach based on samples of the simulated model. One approach, which has been successfully used in a small but challenging problem, is the estimation of a simpler meta-model 47. It is not yet clear whether one empirical model structure is best, and this is an open problem within the PSE, statistics, and machine-learning communities. Identifiability In practice, strong parameter correlation, which might result from insufficient or noisy data or from inappropriate model parameterization, often prevents the unambiguous determination of parameter values 48-50. In this context, one needs to distinguish between structural and practical identifiability. Structural identifiability is a theoretical property of the model structure that depends only on the

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system dynamics and the choice of the input and output variables 51. Practical identifiability is intimately related to the experimental data and the experimental noise 52. Accordingly, there are two main ways of handling non-identifiability. In special cases, structural nonidentifiability can be removed analytically by re-parameterization and the introduction of new parameters that represent an identifiable combination of several non-identifiable parameters 53. However, in the case of knowledge-driven models, these new parameters might not have a physical meaning anymore. This approach is also referred to as a priori identifiability analysis, as the model structure is examined before the parameter-fitting step. In contrast, practical non-identifiability is detected through the fitting procedure by investigating the confidence regions 54, a task related to Question A. Hence, this approach is also labeled a posteriori identifiability analysis. Practical identifiability can be analyzed using the Fisher Information Matrix (FIM) or sensitivity functions 55. The FIM approach emphasizes possible correlation between some of the parameters, while the sensitivity analysis provides information over time that indicates when parameters are poorly or highly identifiable and when some parameters may be correlated, which might be very useful for distributing the available data between calibration and validation, and for preferring some data over others within the same set of experimental data. Furthermore, even if parameter identifiability can be addressed straightforwardly for a specific point of the parameter space, it is more difficult to determine globally over the whole parameter space 56. Also, it would be useful to be able to address model identifiability at an early stage, that is, already at the model building stage, which can significantly contribute to reducing the time needed for identificationoriented experimentation. Can Modeling be Fully Automated? The development of data-driven models strongly depends on the data at hand, and the process is currently widely supported by appropriate software tools. This is much less so for knowledge-driven models, for which therefore the quest for “fully automating” the modeling task would seem quite relevant. A key element regards the steps in the model building process that, currently, may only be captured by a human modeler and not by a computer. A fairly large degree of automation of the modeling task has already been reached in professional process simulators. However, one may argue that most of these modeling platforms have not evolved much from a “unit operations” view of the system to be modeled. On the other hand, Chemical Engineering as a discipline has evolved from the concept of “processes” to that of “unit operations” to that of “transport phenomena”. Hence, one would expect process simulators to evolve similarly toward a transport-phenomena approach to modeling. The modeling of complex systems, including biological components, would benefit significantly from such an evolution. Furthermore, one could envision the modeler assigning the goal of a required model, and the modeling platform returning a model whose complexity is appropriate for that goal (possibly providing a ranking, based on some metric, of different model alternatives). Along the same lines, it would be extremely useful if the model assessment task could be automated (based on experimental data available at the beginning of the modeling task; see Section 4.2) and new experiments suggested for improving model accuracy. According to this vision, a process simulator

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should be able to automatically not only identify and solve the sets of equations describing a given system, but also perform three additional tasks, namely, i) adjust the model complexity to that required by the assigned goal; ii) diagnose the model performance based on the available experimental data; and iii) suggest a set of new experiments to be carried out to improve the model.

4.2. Measured Data and Parameter Estimation Experiment Design Experiment design is central for linking models and experiments 18, 22. Although the field has received much attention over the years, there are still open issues not addressed by existing statistical methods. When a new process or phenomenon is under consideration, the initial set of experiments should aim at discovering which of the many factors have a significant effect. Once this task has been completed, additional experiments are designed in the region of expected improved operation. Finally, a more detailed set of experiments might be designed close to the optimum. Note that one will design sets of experiments differently if they are to be performed in bench-scale or pilot plants as they might require different amount of time, effort and materials and might entail different safety considerations. Still different considerations are applicable if the experiments are to be performed in a production plant, where safety issues are paramount and with the added constraint that the product produced during each experiment must meet customer quality requirements. Note that safety issues are particularly critical when a physiological model is to be used in the context of personalized health care. One way to couple optimality with feasibility and safety is to take parametric uncertainty into account by using appropriate backoffs from the nominal constraints. These backoffs will then be used in the design of experiments to keep the system within a feasible region57. An important open research question concerns the design of experiments that strictly enforce safety and quality constraints without any prior quantitative model describing how the operational conditions affect these constraints. Model Uncertainty and Measurement Errors There are many sources of uncertainty in a model, in particular the fact that certain phenomena are poorly known and others have simply been neglected in the modeling step. Invariably, the experimental data used for parameter identification are noisy and often scarce, which prevents the extraction of richer information contained, for example, in signal derivatives. Furthermore, the data might not have been collected in the most appropriate regions of the experimental space. Certain disturbances and errors are best dealt with at their source. For example, to reduce the effect of measurement errors, one could try to have more precise sensors or, if this is not possible, use redundant measurements. Generally speaking, answers to the following questions (clearly related to aforementioned Questions A, B and C) might be quite instructive: 1. Is it possible to assess the accuracy with which the model represents the measured data? When analyzing residuals in nonlinear regression, can we separate the effect of lack of fit from that of pure error? Are the main process variables well represented by the model?

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2. What is the nature of plant-model mismatch? What variables are less well represented? In the case of knowledge-driven models, could a data-driven approach be used to model these lesswell modeled parts? Another issue regards “model diagnosis”, that is, finding the root cause of process-model mismatch that may arise when a knowledge-driven model is challenged by a set of experimental data 58. 3. Are there sufficient measurements? Could additional measurements invalidate the model? 4. Are the sensors adequate, or would additional sensors provide significantly more relevant information? Having an informative set of data points is key to modeling, which is related to the design of experiments and the optimization of a criterion, associated with the Fisher information matrix 18, 22. It is often possible to improve the quality of measured data via gross error detection and data 59 reconciliation . In practice, gross errors can be detected by a combination of rule-based and cyclic solutions to the parameter-identification problem 60. Data reconciliation estimates “best values” of the measured quantities that are consistent with balance equations and as close as possible to the measurements. The solution can be formulated in terms of an optimization problem, implicitly assuming some error distribution. For data reconciliation to be useful, a certain degree of redundancy in the measurements is required. Note that the least-squares method is not necessarily the best method to reconcile data; robust estimators are excellent alternatives, which however requires considerably more tuning. Ideally, a model should provide not only a structure and some parameter values, but also the input/output region that is statistically supported by the available data. For example, if one wants to use the model for optimization, the most appropriate solution would be to solve a stochastic optimization problem, in which the parameters are seen as random variables with a certain distribution. More research is needed along the lines of (i) what type and amount of uncertainty can be tolerated for a given model application (i.e., not all parts need to be modeled equally well), and (ii) how can one deal with structural plant-model mismatch (i.e., when the plant is not in the model set) and non zero-mean measurement errors.. On-line Model Adaptation and Model Maintenance The strength of a model used for process operation is its ability to predict process behavior, which is quite useful in order to anticipate the effects of input changes and disturbances. Since the model is never a complete and true representation of the process over its global range of operation, it is often adapted locally via additional input excitation and parameter re-identification. On-line model adaptation is primarily needed because the process changes due to disturbances and might well move to a new operating region where its behavior will be different. An important issue regarding on-line model adaptation is the dichotomy between exploration and exploitation (the so-called dual control problem). It is called dual because in controlling such a system the controller's objectives are twofold 61: 1. Exploration: Experiment with the system to learn about its behavior so as to control it better in the future. 2. Exploitation: Control the system as well as possible based on the current system knowledge.

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The process needs to be sufficiently excited for proper identification, while the excitation needs to be minimally disruptive and consistent with operational constraints for good process performance. The close interplay between online experimentation and model usage has often been neglected, and more work is clearly needed along these lines. Soft sensors represent one of the most frequent applications of data-driven models, where the models need to be adjusted on-line 62. The corresponding problem is known under the label of soft sensor maintenance and is central to industrial applications 63. Also note that on-line parameter estimation is closely related to state estimation if the model parameters are considered as augmented states with zero dynamics. By exploiting this relationship, and using for example an extended Kalman filter, even nonlinear on-line parameter identification is possible, though tuning is not straightforward 64.

4.3. Model Validation and Model Improvement Structural Model Improvement An important modeling issue regards the way one proceeds when the goodness-of-fit test, related to Question C, is negative. This happens when the unmodeled sum of squares, SSUM is larger than the sum of squares expected from the normal variability of the process, SSE . In this case, the model needs to be enriched with additional components that are capable of appropriately capturing the unmodeled non-random variability, which is related to the difference SSUM − SSE . The challenge is to decide which additional components to add to the current model. For example, is the mismatch caused by the need to consider a different reaction network, different kinetic models, different mixing assumptions or because mass- and heat-transfer limitations were neglected in the current model? One possible solution to this dilemma is the postulation of several second-generation models, with each one adding a component to the original model. These models can then be tested and compared to see which one represents the data most accurately. Although this is a feasible path forward, the hope is that examination of the residuals of the initial model can guide this process insightfully. An alternative, easier and more expedient approach is to represent the unmodeled variability with data-driven components 65, thus leading to a hybrid modeling. The gradual process of trimming out model components and adding new ones has its parallel in datadriven models under the name of step-wise regression 35. However, with knowledge-driven models, the procedure appears to be more laborious and requires a substantial dose of intuition in the modelenriching step and a hefty process insight. A difficult situation can, for example, arise when the choice of different reaction pathways may lead to model structures that are equivalent in terms of complexity. Although this issue has been studied in the literature 55, it remains an open question, mainly because the reference methods are basically well adapted to linear models, while the considered models are intrinsically nonlinear. Model Portability Once a model has been developed at some expense, one would like to be able to use it at a different scale or in a different environment. Examples are numerous and include the scale-up of a recipe from pilot to production 66, the transfer of a manufacturing policy to a different plant, possibly at a different geographical location, the transfer of “personalized” health care from subject A to subject B, the Page 14 of 27


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transfer of a calibration model to another “similar” instrument 67-68. A key issue is then the question of model portability: to which extent can a model of system A be used to describe system B? The issue of model portability involves mainly data-driven models. For knowledge-driven models, portability is usually assumed to not be an issue as long as the physical principles describing systems A and B are the same, and it often reduces to re-estimating the model parameters. The very problem of model portability for data-driven models reduces to the problem of merging two datasets, namely, a large (historical) dataset related to the source system A and a much smaller one (either historical or to be purposely built) related to the target system B. If merging can be done, then the data-driven model developed for system A can be “ported” (i.e., transferred) to system B. Since the projection techniques used for that purpose are very specific to the problem at hand, the open questions include: (i) Is it possible to derive general guidelines to describe when and how data can be transferred?, (ii) when are two systems “similar” (is there any test to assess similarity)?, (iii) how can historical data, including the preprocessing steps, be transferred?, or (iv) how can fresh data be combined with transferred data? Model Sensitivity A model can be used to perform sensitivity studies between several process inputs and outputs 69. Imagine for a moment that we wish to examine the implication that four process inputs have on some outputs. Such a study would allow investigating, in a global sense, the implications of the available degrees of freedom toward process economics and safety, on the one hand 70, and toward process dynamics and controllability/operability, on the other 71. How many points in the 4-D input space do we need to consider for assessing the operational area of most desirable performance and the areas close to unsafe operation? Consider next a large-scale process for which a detailed and computationally demanding model exists. A way of tackling such a complex process could consist in identifying a simpler meta-model, that is, abstracting that knowledge in a more condensed and explicit input-output form. This approach applies the DoE methodology in silico to the detailed model to estimate a nonlinear data-driven meta-model of the process. Such a meta-model then helps quantify the input-output relationships among selected input and output variables. More work is needed to fully exploit the benefit of model sensitivity toward structural model improvement and specific applications such as parameter estimation, control and optimization.

4.4. Implementation Aspects Modeling Complex Systems We tend to call things that we do not understand well “complex”, which simply means that we have not found a good way of describing them. Complex systems are the norm rather than the exception and can be found everywhere, for example subsea processing of petroleum products, biochemical reaction networks, systems that integrate discrete and continuous dynamics with feedback such as failure and alarm systems. An illustrative list of open problems dealing with complex systems includes (i) the simplification of reaction networks and the estimation of kinetic and interaction parameters, (ii) the understanding of emerging behavior, for example going from a simple molecular behavior to a complex phenomenon as seen in crystallization, (iii) the better use of computational tools for image

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processing, distributed computing, and “big data” handling, and (iv) dealing with “unknown” and safety issues. Mathematical modeling of biological systems must cope with difficulties that are rarely present in traditional fields of applied mathematics. These systems are characterized by a large number of components of extreme complexity and with a variety of interactions. Also of concern is the lack of reproducible and consistent experimental data resulting from the complexity of carrying out experimental trials. This might be due to the dynamics of some systems being exceedingly slow, or the presence of critical safety issues, such as the identification of physiological models of diabetes for single individuals. These difficulties may be overcome by models of a new type, termed ensemble models or set-membership models that allow for the parameters and the model structure to vary and thereby describe a population of all models that are consistent with biological knowledge and measured data. The parameters of ensemble models are estimated using Bayesian techniques, and the models are subsequently used to provide probabilistic predictions about the system behavior and its response to changes in operating conditions 72-73. As already mentioned, models are developed for various objectives in mind. These models can also be very different in nature, ranging from very detailed description of molecular behaviors to macroscopic description of observed phenomena. The challenge in multi-scale modeling lies in the integration of these molecular and macroscopic phenomena 74-75. Furthermore, some models are static in nature and are used mostly for process design, while others are dynamic and helpful in real-time operations. Open problems include (i) the design of experiments to determine which scales and physical locations are important (where/when/which experiments are needed), (ii) the use of modeling procedures that are consistent at different levels of detail, (iii) model reduction for multi-scale models, and (iv) the handling of very large-scale computations through the various scales. Another related issue is the question whether a mathematical model is necessary. With the rapid development of data-driven approaches, one may wonder whether models are still necessary. This question is also justified in the light of the saying that “the best model is, in fact, the data” since data accurately represent the effect of both the plant and the noise. Furthermore, with major advances in dealing with “big data”, one can hope to extract all necessary information through data mining. This reasoning, however, has some limits. Firstly, the model must possess extrapolative capability, which calls for a first-principles model. Secondly, and most importantly, the time spent to construct a mathematical model is an excellent learning opportunity, which often comes as an asset to the modeling team. Numerical Solution Methods The most commonly used mathematical formulations of the parameter estimation problem (e.g. leastsquares or maximum likelihood) correspond to mathematical optimization problems in terms of the parameters to be estimated. In general, these optimization problems are nonlinear since the measured quantities are nonlinear functions (, ) of the parameters and the experimental conditions . Overall, the problem to solve can be written as: min (, | , = 1, . . , )

∈

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subject to: = (, ),

= 1, . . ,

where is an appropriate objective function (e.g. corresponding to the least-squares or maximumlikelihood formulations), is the number of experimental data sets, are the measured values in experimental data set , and Θ is the set of acceptable values of the parameters (typically expressed in terms of lower and upper bounds). For knowledge-based models, the functions are usually not available in closed analytical form, but have to be computed via the numerical solution of the model equations for given values of and . Depending on the type of the model under consideration, this may involve the solution of sets of nonlinear algebraic equations (NLEs), mixed differential-algebraic equations (DAEs) or integro-partial differential algebraic equations (IPDAEs). Moreover, most efficient optimization algorithms (e.g. those based on sequential quadratic programming, or interior-point techniques) also require accurate gradient information in terms of the values of the partial derivatives /. These can be computed to the same precision as the quantities themselves via the solution of the sensitivity equations associated with the model equations. In order to evaluate the objective function and its gradients / for given values of , the computation of the and their gradients / has to be performed for each experiment under consideration. For parameter estimations involving complex models and/or large experimental data sets, this results in very significant amounts of computation. Nevertheless, with advances in both computer hardware and optimization algorithms, it is nowadays possible to address realistic problems of industrial significance, such as those listed in Table 1.

Table 1: Applications of parameter estimation in large-scale industrial process models (Source: Process Systems Enterprise Ltd., 2016)

Process being modeled Ethylene-vinyl acetate batch polymerization reactor Tubular catalytic reactor for exothermic partial oxidation Tubular catalytic reactor for Fischer-Tropsch synthesis High-density polyethylene (HDPE) slurry reactor

(number of timevarying quantities)

Number of parameters to be estimated

25,000

60

20

50,000

40

100

100,000

20

20

100,000

40

30

Approximate model size

Number of experimental data sets (steady-state or dynamic)

However, there are still a number of significant outstanding challenges that will need to be addressed, some of which are outlined below:

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Improved computational efficiency: Notwithstanding recent progress in this area, this remains a key priority area. One reason for this is that knowledge-based process models are becoming increasingly detailed, taking advantage of improved understanding and characterization of the underlying physics. It is now common, for example, to employ models that take account of 1-, 2- and sometimes 3dimensional spatial variations, and/or other distributions (e.g. with respect to particle size or polymer molecular weight) described by population balance equations. Moreover, advances in sensor and analytical technology are resulting in higher quantities of more complex experimental measurements, e.g. measurements of molecular weight distributions and degree of branching in polymers. Clearly, parameter estimation can benefit from advances in optimization algorithms leading to converged solutions being obtained in fewer iterations. When the model involves DAEs or IPDAEs, it is often quite efficient to use “all-at-once” simultaneous solution methods based on models that have been fully discretized by orthogonal collocation 76. A more immediate gain can be obtained within the context of existing algorithms by significantly reducing the time taken by each iteration, in particular via the exploitation of distributed computing architectures. For example, in problems with large numbers of experimental sets, the functions (, ), and their gradients / may be evaluated in parallel for multiple experiments . Moreover, for time-dependent models, the evaluation of the gradients / via the solution of the sensitivity equations may itself be parallelized 77-78, something which may have a significant impact on the overall computation time: for complex parameter estimation problems, sensitivity calculations can account for well over 90% of the total computation. Global optimality of parameter estimates: One serious problem posed by many nonlinear parameter estimation problems is the existence of multiple local minima to the optimization problem posed above. The identification of globally optimal parameter estimates is a significant challenge given the number of parameters that need to be estimated, and the underlying size and mathematical complexity of the underlying mathematical model. A variety of approaches have been proposed in the literature for addressing this problem, taking advantage of recent advances in global optimization algorithms. Much of this work 79-81 over the past two decades has been based on the use of deterministic algorithms that guarantee finite convergence to an -optimal global optimum. However, there is still a significant gap of several orders of magnitude between the model sizes that can be handled by these techniques and those required in addressing problems of industrial significance (cf. Table 1). At present, one practical alternative for these large problems appears to be the use of hybrid stochastic/deterministic global optimization algorithms 82 which solve a sequence of local parameter estimation problems from different starting points (i.e. initial guesses for the parameters to be estimated) that are generated via a systematic search procedure, coupled with appropriate stopping criteria. Such “multistart” approaches provide no theoretical guarantee of global optimality within a finite amount of computation. However, in practice they are often capable of producing significantly better solutions than can be achieved via a single local optimization, especially when implemented on distributed computing architectures that allow the evaluation of large numbers of starting points. Nonlinear confidence regions: As has already been mentioned earlier in this paper, the assessment of the accuracy of the parameter values obtained via the parameter estimation calculation is a crucial element of the validation of mathematical models using experimental data. A key tool in this context is provided by the confidence regions. In most current applications of parameter estimation, these are approximated as ellipsoids centered on the optimal parameter estimates. The approximation is derived

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via a Taylor expansion around the optimal estimate, and is readily computed from the parameter covariance matrix. However, for nonlinear problems, the true confidence regions may differ significantly in both shape and size from the above ellipsoidal approximations, which may lead to severely distorted assessments of the quality of the results of parameter estimation. Techniques for addressing this issue, such as those based on the likelihood ratio 83, have been proposed, but to date their use in applications related to process modeling and model-based engineering 84 remains rather limited. Software Environments for Parameter Estimation Although the use of software tools for process modeling is now quite widespread in both industry and academia, only relatively few of these tools support the estimation of model parameters from experimental data. One potential reason for this is that the reliable and efficient solution of the underlying optimization problem requires access to partial derivative information which cannot easily be obtained by tools based on the sequential modular architectures, a category that includes most commonly used steady-state flowsheeting tools. Thus, parameter estimation capabilities are largely limited to tools conforming to the equation-oriented paradigm, such as gPROMS® (Process Systems Enterprise Ltd.) and Aspen Custom Modeler® (Aspen Technology Inc.), as well as Aspen Plus® (Aspen Technology Inc.) when operating in the equation-oriented mode. A second complicating factor is that many laboratory-scale experiments, especially those used for the identification of reaction kinetics, are performed using batch or semi-batch equipment. Only tools that support modeling of transient processes are applicable to such cases. Beyond the ability to address current and future challenges relating to the numerical solution of parameter estimation problems and the statistical analysis of its results, effective support for parameter estimation poses additional demands on process modeling software. Of particular importance is the incorporation of these calculations within an overall workflow of tightly integrated modeling and experimental R&D, rather than viewing them as isolated computational tasks. For example, an efficient and error-free workflow requires that the process modeling software be able to directly access and process experimental measurements produced by standardized experimental procedures and/or analytical instruments. Moreover, non-trivial applications often involve multiple estimation steps applied to different sets of parameters using experimental data from different experimental set ups. For example, the design and optimization of industrial multitubular catalytic reactors for exothermic reactions, such as partial oxidations, can nowadays be performed to a high degree of sophistication using very detailed knowledge-based models that take account of axial, radial and intra-particle variations. Despite their firm foundation on first principles, such models typically require at least three sets of parameters that cannot be fully predicted theoretically and therefore need to be estimated from experimental data; these relate, respectively, to (a) the geometric characteristics (e.g. tortuosity, pore size distribution etc.) of the catalyst pellets (b) the catalytic reaction kinetics, and (c) the thermal conductivity of the packed bed and the bed-to-wall heat transfer coefficient. Best practice dictates that a separate type of experiment be employed for characterizing each of these three physical phenomena, ideally in a manner that decouples it to the maximum extent possible from the others. For example, the pellet characteristics can be determined via standard porosimetry techniques independently of the other

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parameters; the identification of the kinetic parameters may be done based on experiments performed in a small-diameter tube packed with diluted, often pulverized, catalyst and operated under near-isothermal conditions, thereby eliminating any dependence on the heat transfer parameters. The latter may then be identified via experiments using a single full-size tube packed with standard catalyst pellets and surrounded with a well-controlled and well-characterized cooling jacket. The model used for this last parameter estimation step requires as inputs both the pellet characteristics and the kinetics determined by the earlier parameter estimation stages. Finally, all three parameter sets become inputs to the model used for optimizing reactor design and operation of the full-scale multitubular reactor. Thus, the software architecture of process modeling tools needs to support the efficient, automated and error-free transmission of information among all these complex computational steps. Industrial View of Modeling Despite much research activity, there still exist numerous technical and non-technical barriers that hinder a wider introduction of models into industrial practice. These barriers include: 1. Technical gaps, such as the lack of appropriate model and simulation capabilities for complex processes, the lack of validated thermodynamic and kinetic data, and the lack of a high-level process synthesis methodology. Both the development of a model and its subsequent use, for example for model-based control applications, typically require implementation of the candidate model in a suitable software environment. Unfortunately, modeling software does not always offer all required functionalities, and specialized software may not be available due to licensing costs, platform dependence, unsupported model type or limited user experience. While some of these issues can be resolved by using open-source modeling environments and languages, such as Modelica 85, transferring a model from one software environment to another remains a significant challenge. In industry, initial attempts to establish a standardized interface for model exchange have been made with the Functional Mock-up Interface (FMI) 71, but more research is needed until all model-related properties, such as parameter sensitivities, can be exchanged with minimal user interaction 2. Technology transfer barriers, such as the lack of multi-disciplinary team approaches in process integration, the lack of problem commonality (technology is very application specific) and the lack of demonstrators/prototypes on a reasonable scale, the language and communication barriers. With respect to the “language barrier”, people with different backgrounds often use the same words to indicate different things. A classical example is the U.S. Food and Drug Administration documents on Process Analytical Technology and Quality by Design, where the term “control” is used with a totally different meaning from what engineers are used to. Regarding the communication barrier, it is often difficult to communicate the results of a given development to the lower technical levels in such a way that the project outcome can bring changes to industrial practice. 3. General barriers, such as the higher standards to which new technologies must be held compared to conventional technologies, the lack of information on process economics (early economic and process evaluation) and the fear of risk in using new technologies. Most of these barriers can definitely be overcome, and universities have here a fundamental role to play, not only by developing new catalysts, new materials, new methodology and tools for high-level

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process synthesis, but also by teaching future chemical engineers an integrated, task-oriented approach to plant design 86. At this point, it is interesting to mention two important issues that future research and educational innovations need to address. The first issue concerns the gap between what is done in industry and what is done in academia. It is difficult to quantify the magnitude and character of this gap, because industry does not communicate extensively about the type of models that it develops and uses. The definition of industrially relevant benchmark problems would be most welcome. This will greatly contribute to significantly boost the research on modeling, similarly to the impact that the Tennessee Eastman Process Control Problem 87 has had on the control community. The second issue is linked to education and concerns the gap between what undergraduate curricula teach (or fail to teach) about modeling and what is needed and practiced in industry.

5. Conclusions Considering that the data collected on a process constitute in fact the “best model”, it is not surprising that models and experiments are intimately linked. Consequently, it is important for model building to utilize data appropriately and in a statistical framework because of a multitude of uncertainties, including the presence of measurement errors. In any case, model validation is crucial and must be performed using sound statistical tools. Modeling is not a religion, and there is no such thing as the “true model”. This paper has shown that the models can be of different classes, different types and different structures and can therefore be tailored to cover most needs. Whether a given model is useful or not depends on the application at hand. Simulation, control and optimization typically require models with specific (i.e. different) attributes. Unfortunately, the obvious concept of “modeling for a goal” does not seem to have percolated fully into the research community. As a result, the modeler often spends most of his/her time trying to make the model fit the available data (most often process outputs) instead of investigating what the best data should be for a given application! Since the model must be tailored to its intended purpose, it may be useful to test the model suitability with respect to the planned application. This paper has identified several research directions that can help develop and consolidate a systematic and sound modeling methodology. If accepted by the majority of researchers, such a modeling framework would provide reasonable assurance that models published in the literature and models used in industry are indeed statistically correct.

References 1. Ljung, L.; Glad, T., Modeling of dynamic systems. PTR Prentice Hall: Englewood Cliffs, N.J., 1994; p 361. 2. Rasmuson, A.; Andersson, B.; Olsson, L.; Andersson, R., Mathematical Modeling in Chemical Engineering. Cambridge University Press: 2014; p x, 183 pages. 3. Hangos, K. M.; Cameron, I. T., Process modelling and model analysis. Academic Press: San Diego, 2001; Vol. 4 of Process Systems Engineering, p xvi, 543 p.

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Figure

Figure 1: Number of publications with “Modeling” in the title, 1975—2014.

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Linking Models and Experiments - Industrial & Engineering

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