Hybrid Modeling Approach Integrating First-Principles Models with

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Process Systems Engineering

A hybrid modeling approach integrating first principles models with subspace identification Debanjan Ghosh, Emma Hermonat, Prashant Mhaskar, Spencer Snowling, and Rajeev Goel Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b00900 • Publication Date (Web): 28 May 2019 Downloaded from http://pubs.acs.org on June 2, 2019

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Industrial & Engineering Chemistry Research

A hybrid modeling approach integrating rst principles models with subspace identication †

Debanjan Ghosh,

†

Emma Hermonat,

∗,†

Prashant Mhaskar,

‡

Spencer Snowling,

‡

and Rajeev Goel

†Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada,

L8S 4L7 ‡Hydromantis Environmental Software Solutions, Inc. 407 King Street W., ON, Hamilton,

L8P 1B5 E-mail: [email protected]

Abstract This paper addresses the problem of synergizing rst principles models with datadriven models. This is achieved by building a hybrid model where the subspace model identication algorithm is used to create a model for the residuals (mismatch in the outputs generated by the rst principles model and the plant output) rather than being used to create a dynamic model for the process outputs. A continuous stirred tank reactor (CSTR) setup is used to illustrate the proposed approach on a continuous system. To further evaluate its ecacy, the proposed methodology is applied on a batch polymethyl methacrylate (PMMA) polymerization reactor and the predictions are compared with that of rst principles modeling and the data-driven approach alone. The paper demonstrates the improved modeling capability of the hybrid model over either of its components.

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Introduction

Good models are integral to process systems engineering, often being the core around which control and optimization formulations are designed. First principles or mechanistic modeling refers to the type of modeling where explicit knowledge of the process mechanism is present and utilized.

Thus, these models invoke fundamental physical and chemical laws that

describe the system being considered and utilize algebraic, ordinary or partial dierential equations. Often times, the fewer the assumptions made, the higher the number of model parameters and the more complex the model structure. The task of building and maintaining a rst principles model 17 continues to involve signicant eort. Increased availability of data, along with improved computational capabilities have made the so-called data-driven methods increasingly attractive 810 . In this approach, a model structure is chosen a priori and parameters are identied using the available data. In this direction, several statistical based approaches exist and are distinguished by the kinds of model structures being utilized, including latent variable-based methods and subspace identication-based algorithms. Projection to latent structures (PLS) is one of the most recognized statistical modeling methodologies and has shown signicant advantages over its counterparts. PLS is one of the latent variable methods where measurement data of high dimension is projected into lower dimensional space to create simpler and eective models. Their usefulness to batch processes have been extensively demonstrated, 9,11 with special algorithms utilized to enable handling of batches of non-uniform lengths. In this method, while process knowledge is often integrated into the modeling approach by utilizing additional calculated variables, a direct integration of rst principles models and latent variables models has not been done. Another statistical-based modeling approach is subspace identication, which enables the identication of models having state-space representations. In these methods, a state trajectory is rst identied and the model parameters (for a linear time-invariant model (LTI)) are then determined (see 8,1215 for examples of subspace identication methods). 2


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Recently, subspace identication-based methods for modeling batch processes 1619 have been proposed. A subspace identication scheme where the batch lengths do not necessarily have to be of the same size was proposed in 16 and its merits in prediction and control were demonstrated by applying it to a batch polymerization process. In addition, a modeling strategy was proposed in 17 that allows discrete addition during the batch. The usefulness of this identication scheme was shown in 18,19 by applying it to a batch particulate process and a hydrogen plant start-up process. The approaches described above, however, have not demonstrated the utilization of available process knowledge in synergy with the data-driven modeling approach. There have been some examples in the past of approaches that have utilized a priori knowledge in combination with articial neural networks (ANNs; see 2023 ) to create hybrid, or grey-box, models. Readers are encouraged to go through 24 , where an excellent overview of dierent hybrid semi-parametric modeling approaches is provided. The various applications of these models, such as process monitoring, optimization and control in the eld of process systems engineering are also discussed. It should be noted that in this work, hybrid semi-parametric models are referred to as simply hybrid models. Hybrid modeling approaches can be broadly categorized as either series (where the models are used one after the other) or parallel (where the models are used in parallel to one another). Examples of the series approach are provided in 3,25,26 where the hybrid model is built by rst by developing the rst principles model and neural networks are subsequently used to determine the parameters. The approach was later demonstrated in 3 with uncertain, unmeasurable and unknown parameters. In the parallel approach, a data-driven modeling structure is utilized to model the error between process outputs and the predicted outputs from the rst principles model (referred to as residuals). Typically, a neural network model is utilized as the data-driven modeling scheme 27,28 ; however, the utility of ANN-based models to capture the process dynamics while avoiding the problem of overtting continues to be an active research focus. On the other hand, statistical approaches such as subspace identication-based models 3



permit a more direct handling of the overtting problem. Therefore, the proposed approach follows a parallel hybrid modeling scheme and uses subspace identication as the data-driven component. A key advantage to the hybrid modeling approach is that it oers an alternative approach to addressing the issue of model maintenance that is associated with using rst principles models. As process conditions change and new data becomes available, a recalibration of the rst principles model (re-identifying the model parameters from process data) will likely result in a more informative model. As stated earlier, the recalibration process is limited by the diculties associated with solving a highly complex optimization problem. The hybrid modeling approach instead exploits the fact that even though the rst principles model may no longer be able to predict the variables precisely, it still accounts for the qualitative nature of the (often nonlinear) process dynamics. Therefore, building a simpler data-driven model that captures the residuals negates the need for model recalibration or re-identication given that the hybrid model accounts for both the process nonlinearity, and the eect of the new process conditions. In summary, while the ability of the subspace algorithm for modeling batch processes has been established 1618 , this eective modeling tool has not yet been integrated with rst principles model in a hybrid modeling framework. Motivated by these considerations, a hybrid model integrating rst principles modeling with subspace identication algorithm is proposed in this work. The rest of manuscript is organized as follows: Section 2.1 presents a PMMA batch process as a motivating example and a brief overview of the subspace identication method employed is given in Section 2.2. Section 3 describes the development of the proposed methodology. The hybrid approach is rst illustrated using a CSTR case study in Section 3.4. The methodology is also applied to a PMMA polymerization process and compared with the existing subspace identication approach in Section 4. Finally, concluding remarks are presented in Section 5.

4


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2

Preliminaries

In this section, we briey review a PMMA polymerization process example to motivate the proposed research. Given that the model in the proposed methodology considers subspace methods as the identication tool, a brief review of subspace identication is presented in the following subsection.

2.1 Motivating example: PMMA polymerization reactor A batch polymerization of PMMA in a stirred tank reactor is used as the motivating example in this study. The ingredients of this recipe are the monomer methyl methacrylate, the initiator AIBN and toluene as solvent. The reactor temperature is maintained using a cooling/heating jacket. To simulate the process, the mechanistic model presented in 29 , with alterations taken from 5,30 is utilized (see 31 for further details). A total of nine states are present in this mechanistic model, which includes the concentration of monomer, the concentration of initiator, the reaction temperature and six moments of living and dead polymer chains. The model with the associated parameters is considered as the 'process' in the present manuscript. To replicate practice, data is generated for the batches using variation in the initial conditions. The measured outputs for this polymerization process are chosen to be the reaction temperature, log viscosity, and density. The manipulated input variable is the jacket temperature (Tjk ). The process dynamics display signicant nonlinearity and are highly dependent on initial conditions, making it important for the hybrid modeling approach to be implemented in a way that estimates current process states before being used for prediction purposes. For the purpose of model identication, it is thus assumed that a total of NT training batches and NV validation batches are available with Db , b = 1 . . . , NT and Dv , v = 1, . . . , NV being the duration of training and validation batches respectively. Note that the training and

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validation batches are not required to be of uniform length. The input (jacket temperature) is denoted by u = Tjk and the measured outputs (reaction temperature, log viscosity, and T density) are represented by the column vector y = Treac log µ ρ . It is assumed that a rst principles model for the process is available, albeit with a mismatch between the process and model. Note that using existing results, a subspace model between the input and outputs can be readily determined. This paper specically addresses the problem of developing a hybrid model that combines the rst principles and data-driven models. Before proceeding to present the proposed approach, the subspace identication method is briey reviewed.

2.2 Subspace identication Subspace identication techniques are methods which use matrix decomposition concepts of linear algebra such as SVD (singular value decomposition) and QR factorization for the identication of a discrete time linear time-invariant state-space model of the form below:

xsd [k + 1] = Axsd [k] + Busd [k]

(1)

ysd [k] = Cxsd [k] + Dusd [k] where xsd [k] denote the subspace states of the system at a time instant k and is a (n × 1) vector. The subspace states of the system are abstract in nature and may not directly relate to the states described by the rst principles model. The number of states, nsd , is the order of the identied system and is also one of the parameters of the identication procedure.

ysd [k] and usd [k] are (lsd × 1) and (m × 1) vectors respectively and denote the predicted outputs and manipulated inputs. For a system of a particular order, the model parameters to be computed include the matrices A, B, C and D, and the initial condition of the state variable for the training data set. In this approach, a state trajectory is rst identied and the system matrices are later computed using linear regression. These methods are non-iterative in nature, as opposed to 6


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prediction error minimization (PEM) algorithms, which involve estimation using linear and nonlinear optimization problems. Some of the widely used subspace algorithms are canonical variate analysis 32 (CVA), numerical algorithm for subspace state-space identication 33 (N4SID) and multivariate output error state-space 34 (MOESP). These methods were shown to be the interpretation of an unied framework in, 35 with the dierent methods resulting from decomposition of weighing matrix chosen dierently in each case. The subspace identication algorithm presented in 36 has been used in the present work.

3

Proposed modeling approach

While other instances of hybrid or grey-box models exist in the literature 2628,37,38 , the specic architecture proposed for the hybrid dynamic model is key to achieving the objective of enhancing the rst principles models. In the present section, the hybrid modeling approach is described for the more challenging case of batch data (including data from multiple batches of non-uniform length). The rst simulation example illustrates the approach using the simpler case of a continuous CSTR followed by the application to the motivating batch PMMA polymerization example.

3.1 Model identication Model identication is the process of using measured data to build a mathematical model of a dynamic system. Following identication, model validation techniques are used to verify that the estimated model accurately reproduces the dynamic behaviour of the system. When identifying a dynamic model, both the model parameters and the initial state of the system are identied. This is true for dynamic models in general, regardless of the technique used, whether that be subspace identication or other techniques such as neural networks. For a new data set, the model simply cannot predict (accurately) until the states of the system have been estimated using a portion of the new data. In contrast, when developing static 7



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models, once a model has been developed/trained, it can be directly implemented for new data to predict the modeled variable. To account for this feature of dynamic models, the identication step is described rst, followed by the validation step. Recall that a total of NT training batches and NV validation batches are available, with

Db , b = 1 . . . , NT and Dv , v = 1, . . . , NV being the duration of the training and validation batches, respectively. Thus, for the bth batch, input and output data of the form [uk , yk ]b is available. It is assumed that a rst principles model for the process is available, albeit with mismatch between the process and model. Note that using existing results, a subspace model between the input and outputs can be readily determined. This paper specically addresses the problem of developing a hybrid model that combines the rst principles and data-driven models.

3.1.1 First principles model The rst step in this modeling approach is to create the rst principles model using available process knowledge. Note that the key contribution of the proposed approach is not the development of a rst principles model, but the utilization of the rst principles model within a hybrid modeling framework. To this end, we assume the existence of a rst principles model of the form:

x˙ f p = f (xf p , uf p )

(2)

yf p = g(xf p , uf p ) where xf p ∈ Rnf p denote the nf p rst principles model states, and f and g are functions (possibly nonlinear) which determine the evolution of the states and outputs. yf p ∈ Rlf p are the predicted outputs and uf p ∈ Rnm are the inputs. lf p and nm denote the number of measured outputs and manipulated inputs and are considered the same as the process. As the rst step of the hybrid model identication procedure, the predictions for all the batches are generated using known initial conditions of the rst principles model and the 8


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known input values for the batches. Thus uf p are chosen to be the same as u (inputs to the process) to compute the rst principles model prediction. The predicted values of the (b)

states and outputs at the k th instant for any batch b are respectively denoted by xf p [k] and (b)

yf p [k].

Remark 1 We assume the availability of the initial states (x

0 f p)

of the rst principles model

for all the batches. In general, we may not have measurements of all the initial state values. In such situations, the rst principles model would need to be run with an appropriate state estimator rst and then be available for prediction after the outputs converge. This is not a limitation of the present work and would be required even in the absence of the hybrid modeling. In other words, if the rst principles model is such that the initial states of the model are not available for measurement, the implementation of the rst principles model would require the state estimation setup. Thus, the proposed hybrid modeling framework does not impose any additional requirements. An explicit illustration of this scenario is a subject of future work.

Remark 2 In deciding whether or not the rst principles models alone, the data-driven model alone or the hybrid model alone should be utilized, the following guideline is suggested: For the cross-validation batches, compute the Mean Absolute Scaled Error (MASE) under the rst principles model and the best data-driven model only (where the data-driven model is directly modeling the process inputs and outputs). Next, using the number of states in the data-driven model as the upper bound, determine the best hybrid model. The model that yields lowest MASE should be the one chosen. The expected outcomes of this exercise are one of the two: Either the hybrid model is chosen or the data-driven model alone is chosen, with the outcomes dependent on the predictive capability of the rst principles model. In particular, if the rst principles model is not suciently quantitatively informative, the data-driven model will typically outperform a poor rst principles model and be chosen. Note that such an outcome could also be used as a trigger to re-calibrate the rst principles model. On the other hand, if the rst principles model has useful information, then the hybrid model is 9



the expected chosen outcome, with the residual data-driven model improving upon the rst principle model.

Remark 3 The rst principles model does not necessarily have to be in the form of an explicitly written dierential algebraic equations (DAE) set. Instead, it could very well be a complex industrial simulator which combines several interconnected units to generate the outputs. The only requirement is that the inputs and outputs should be the same as those of the plant. Again, this is not a limitation imposed by the proposed approach, but would need to be in place for the rst principles model, to begin with. Finally, for the rst principles model to be useful in real time for control purposes, it should be possible for the simulation for each step to be completed signicantly faster than the sampling time between measurements.

3.1.2 Subspace-based residual model Consider now the scenario where plant data (y, u) for training and validation is available and the rst principles predictions (yf p ) have been generated according to Section 3.1.1. This section illustrates the development of the model between the residuals e = y − yf p and the input. Figure 1 shows the layout of the model building step. u

u

Process

First principles model

u

u y-yfp

Subspace Identification

y

yfp

A,B,C,D

Figure 1: A schematic illustrating the integrated modeling approach. The top schematic shows the input-output data collected from the process. The same input is fed to the rst principles model to generate the respective outputs. Finally, a model is built between the error and process inputs. The identied model is a discrete LTI sytem of the form: 10


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xhm [k + 1] = Ahm xhm [k] + Bhm uhm [k]

(3)

ehm [k] = Chm xhm [k] + Dhm uhm [k] where xhm ∈ Rnhm denote the nhm subspace identied residual model states. Ahm , Bhm , Chm and Dhm are the system matrices. ehm ∈ Rlhm is the vector of residuals where lhm denotes the size of the vector and is the same as lf p . The system matrices are computed by utilizing the batch subspace identication outlined below. uhm is the input to the model and is same as uf p . One of the key steps in subspace identication is the arrangement of input-output data to create appropriate Hankel matrices to be used in the algorithm. Instead of using the outputs, in the proposed approach, the Hankel matrix of residuals for the bth batch is formed as shown below:

(b)

E1|i

  (b) (b) (b) (b) e [2] ··· e [j ] e [1]    . . .  . .. .. =  .    (b) (b) (b) (b) e [i] e [i + 1] · · · e [i + j − 1]

(4)

A total of 1, 2, . . . , i, . . . , i + j − 1 continuous measurements are used to build the matrix. The subscript 1|i in this matrix refers to the aspect that elements 1 to i is present in the rst column of the Hankel matrix. To appropriately utilize data from multiple batches, the data from each batch is concatenated 16 to create a Hankel like matrix, given by:

(2) (NT ) E1|i = E(1) 1|i E1|i · · · E1|i

(5)

with NT being the total number of batches used for training the model. (b)

The input matrix U1|i is created in a similar manner. The value of i is chosen to be slightly greater than the expected number of subspace states. This modeling approach can

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naturally handle batches of non-uniform length. Note that the index value i is the same for all batches to create a matrix that has an equal number of rows for all the batches. However, the number of columns can vary depending on the duration of the batches. From Equation 3, through repeated substitutions, the following equation is obtained: (6)

Eh = Γi .X + Ht .Uh

where, Eh and Uh are the output and input Hankel matrices, Γi is the extended observability matrix, X is the consecutive state vector sequence and Ht is the lower triangular block Toeplitz matrix. Two matrices H1 and H2 are constructed via the concatenation of E1|i ,

U1|i and Ei+1|2i , Ui+1|2i respectively, as shown in Equation 7. The subscript i + 1|2i indicates that elements indexed from i+1 to 2i are arranged in the rst column of that Hankel matrix.







 E1|i  H1 =  ; U1|i



 Ei+1|2i  H2 =   Ui+1|2i

(7)

Next, matrix H is constructed as the concatenation of H1 and H2 as shown below:   H1  H=  H2 The state vector is realized as the intersection of the row space of the two block Hankel matrices H1 and H2 . Therefore, the state sequence vector can be obtained by performing an SVD of the matrix H, and is given by X2 = [ xhm [i + 1] xhm [i + 2] · · · xhm [i + j] ]. Subsequently, the system matrices are computed using least squares of the over determined set of equations shown below:





    xhm [i + j]  Ahm Bhm  xhm [i + 1] · · · xhm [i + j − 1] xhm [i + 2] · · · = .  (8)  ehm [i + 1] · · · ehm [i + j − 1] Chm Dhm uhm [i + 1] · · · uhm [i + j − 1]

12


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Alternative approaches exists in literature 34 , where the system matrices are computed directly from the I/O data without the requirement of rst computing the state sequence. In this subspace-based model building methodology, nhm and i are the parameters that need to be specied and thus have to be chosen adequately for the model to provide good predictions. Both nhm and i can be determined by using cross validation, or i chosen heuristically (while satisfying i > nhm , and for the present application, i = nhm + 4. In the existing batch subspace identication 16,17 results, these parameters were chosen by trial and error. In the present results, a cross validation technique for determining the best number of states is utilized and presented in Section 3.3. Having picked the best values of nhm using cross validation, the entire training data set is utilized to determine the system matrices. Before presenting the cross validation method, however, the model validation approach is presented in the next section to enable a clearer understanding of cross validation.

Remark 4 One of the key diculties with the utilization of rst principles models is known to be the model maintenance aspect, more so than the model development aspect. In other words, while it may be possible to put in the resources to develop a detailed rst principles model, practitioners struggle with the resources required to update or maintain the model as process conditions change or new data becomes available. The proposed approach is developed to address model maintenance; therefore the model is not recalibrated as new data becomes available (i.e. the parameters of the original rst principles model are not re-identied). Instead, the new data is used to 'correct' the error between the existing rst principles model and the process data, thus requiring the development of a model that captures the residual.

Remark 5 It is important to recognize that the hybrid model does not try simply to model the residuals as a stochastic process. If one were to do that, one would lose the opportunity of recognizing (and quantifying) that the discrepancy between the rst principles model and the process could be impacted by the inputs themselves. The proposed hybrid modeling approach builds a dynamic model between the inputs and the residuals between the process outputs and the rst principles predictions, allowing the residuals to be predicted forward. Using this 13



structure, the hybrid model is able to capture much of the nonlinear process behaviour that is accessible through the rst principles model as well as the remaining (often almost linear) dynamics that is captured by the subspace identication-based model. Thus, the structure of the model chosen in the proposed hybrid model framework is key to the success of the proposed hybrid approach.

Remark 6 Note that a hybrid model could very well include the rst principles model and an empirical nonlinear model. It is generally expected that the resultant hybrid model would still perform better than a purely rst principles or purely empirical nonlinear model. The concern of overtting with nonlinear empirical models, however, continues to be a research challenge. In contrast, the subspace identication-based approaches generally tend to not overt, and the `information' captured through additional states can be readily inferred from inspecting the SVD of the block Hankel matrices, and utilizing cross validation, as illustrated in the manuscript.

Remark 7 For continuous processes, the data sets required for building the model can be generated by operating the process around a desired operating point. However, in batch processes, the operation is typically over signicantly varied dynamics. Thus, a model which captures the varying dynamics of the process over the entire range is required. Since dierent batches have varying initial conditions, a model built with data from a single batch would be inept to make desired predictions and, therefore, necessitates the use of training data sets collected from multiple batches. The use of data from multiple batches, especially in a way that recognizes that the endpoint of one batch is not the starting point of another batch is important and is utilized in the proposed approach.

3.2 Model validation For a fresh batch of data, the model can be used for the purpose of prediction only after the states of the residual model have been adequately computed. A state estimator, denoted by 14


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¯ hm and generate SEhm (e, u), is used initially to estimate the subspace state vector sequence x ¯hm . The estimated output of the hybrid framework is given by y ¯ = yf p + e ¯hm . the outputs e ¯ ) converges to the actual process output (y ), after The estimator is used until the output (y which the hybrid model is ready to predict. From this time point on, the converged state is considered to be the initial state of the identied state-space model for prediction. The model is then used to predict the output ehm using future values of inputs (u). ypred = ehm + yf p is the predicted output of the hybrid model. The above model validation procedure is schematically shown in Figure 2. No

Identified model (Ahm,Bhm,Chm,Dhm, nhm) + Observer (y,uhm,x0hm)

ekhm

Calculate output, = ykfp + ekhm

Calculate prediction error, epred = y-

At k = k*, Is epred < tol

Yes

Using the current state as the initial state for model Yes prediction, xkhm = khm For k = k*+ 1 to end

Final prediction, ykpred = ekhm + ykfp

xk+1hm= Ahmxkhm + Bhm uhm ekhm= Chmxkhm + Dhmuhm ykfp

First principles model, ykfp

k

Using values of y

fp

(k = k*+ 1 to end)

Figure 2: A schematic illustrating the validation procedure Thus, the validation sequence consists of rst a portion where the outputs from the validation batch are being utilized for state estimation (as shown in Figure 3(a)) and the prediction stage where the outputs are being predicted using only the future inputs and the state estimated during the state estimation step (as shown in Figure 3(b)).

Remark 8 In this work, a Kalman lter is used for state estimation. The error tolerance value considered for convergence using the Kalman lter is chosen be a user specied value in the two case studies. However, it can be set by nding the average standard deviation values of the error trajectories of the training batches. This method, however, is not restricted to this particular choice of the state estimator and any other suitable state estimation scheme can be utilized as a part of the hybrid model. 15



uFirst Principles model

u

yfp

u +

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yfp +

+

ypred

+

Subspace model

Observer

epred

e=y-yfp (a)

(b)

Figure 3: A schematic illustrating the model validation procedure, where (a) shows how the measured outputs and inputs from the present data are used to compute the state estimate and (b) depicts how the model can be used for prediction purposes.

Remark 9 The ability of the model to be able to predict (without requiring measured outputs and observer) is of key importance to their potential use in advanced control strategies. In particular, for instance, at a particular time step, a model predictive control (MPC) formulation utilizes the model and candidate future inputs to predict future outputs and evaluate objective functions. It is this predictive capability of the model that imbues the MPC with all the optimality benets. Thus, for cross validation purposes, the model's predictive capability is evaluated, not just the ability of the state estimator to estimate outputs and states using measured outputs.

3.3 Cross validation methodology The cross validation approach to determine the order of the subspace based model is as follows. The batches used for model building are rst divided randomly into k groups or partitions. For the purpose of cross validation, these k groups are divided into training groups, which are collectively represented as CVtraining , and the validation or testing group is represented as CVtest . For the rst run, the rst group is chosen to be the validation set (CVtest ) and the rest of the k − 1 groups constitute the training set (CVtraining ). With a particular choice of the number of states in the identied model, nhm , the model is built using the batches present in the training set. The model is then tested/validated on the 16


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validation set to evaluate its predictive capability. The prediction error is quantied by calculating the mean absolute scaled error of the square error (MASE) between the model prediction and the plant output data and is given by MASE =

T T −1

PT |et | PTt=1 t=2 |Yt −Yt−1 |

where et = Yt − Ytpred , Yt and Ytpred being the process output

and the predicted output respectively at the t th sampling instant and T is the range over which the error is calculated. This procedure is repeated k times so that each group has acted as the validation set once. The cumulative error value is then calculated for that particular state value. Once this is done, the number of states is increased by one and all the steps are repeated and the nal error value recorded. Finally, the model order (number of subspace states) is chosen as the one that yields the least error during the process.

3.4 Simulation results for a CSTR Consider the case where a rst order exothermic irreversible reaction A

B takes place in

a CSTR that is surrounded by a jacket. The rst principles model is given by the equations below and the parameter values are listed in Table 1.

−E F dCA = (CA,in − Ca ) − k0 exp CA dt V RT F ∆H dTreac −E Q = (Tin − Treac ) − k0 exp CA + dt V ρCp RTreac ρCp V

Table 1: CSTR parameters Parameter nomenclature CA,in : Concentration of A in feed stream ∆E : Activation energy CA : Concentration of A in reactor V : Reactor volume ∆H : Heat of reaction ρ: Density Cp : Heat capacity Q: Heat removal/addition k0 : Pre-exponential factor F : Volumetric owrate R: Ideal gas constant Treac : Temperature in the reactor Tin : Inlet Feed temperature

17


(9)


The CSTR setup as described above is utilized as a test bed for the simulation study. All process data is generated by using the above set of dierential equations for the CSTR. The inputs considered are the heat removal/addition (Q) and inlet concentration of species A (CA,in ). The inputs are constrained between bounds (Table 5) and their initial value is randomly chosen from a uniform distribution between the boundary values. They are implemented in a way such that they are held constant over a period of 50 time steps, where each step is one minute. The measured outputs are the concentration of species A (CA ) in the reactor and reaction temperature (Treac ). The initial values of CA and the Treac are randomly chosen each time from a normal distribution with a standard deviation value provided in Table 6. Plant output data is considered to be corrupted by measurement noise. The noise is assumed to be Gaussian with zero mean and the individual standard deviation value are listed in Table 6. A moving average lter is used to attenuate the eect of noise on the output signals. The process is run for 1000 minutes with data sampled every one minute to generate 12 sets of data. We also assume that we know a model of the process, albeit with error to replicate a real life scenario. The rst principles model has the same model structure as the process model, except with 5% deviation in its parameters (5% for ∆E and -5% for k0 ). The parameter values considered for this case study are listed in Table 4. The training and cross validation is carried out as described in the previous sections. The cross validation approach is implemented to nd the best choice of states for both the hybrid and subspace case. To this end, the 12 data sets are randomly divided into four groups as shown in Figure 6(a). An initial choice of 7 Hankel rows is considered for both the methods. As explained in the previous Section 3.3, the model is trained on three groups and validated on the remaining one group. This training/validation method is carried out until each of the four groups have acted once as the testing set. Model validation is then carried out as per the steps explained in Section 3.2. A Kalman lter is used to estimate the states using the knowledge of system matrices, the output measurements, and noise data. The measurement noise data is listed 18


Page 18 of 32

Page 19 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


in Table 6. The initial state value of the Kalman lter is considered to be a zero vector. The model is used to predict for the rest of the batch once the outputs have converged. For the validation batches, the outputs converge at dierent time instances. In order to have a fair comparison, all the error values in the dierent simulation runs were computed after the rst 250 time instances. The cross validation step yields 3 as the best choice of subspace states. The validation results for a fresh run are shown in Figure 4 and the prediction error values are listed in Table 2. In order to correctly judge the quality of the model, only the model prediction part of the two methods is considered for comparative analysis. The Kalman lter is used till the outputs for both the methods converged and is kept in closed loop up to k = 250 th instant. The error values for the model prediction part are calculated from that point onward to the end of the batch. There are a number of observations to be made from the validation results in Figure 4. The rst is that the rst principles model, while not predicting accurately, still captures the general nature of the process, thus making it a useful component of the hybrid model. This is corroborated by the fact that the results from the hybrid model predictions are better when compared to the predictions using a classical subspace model alone. In summary, it can be seen that the hybrid model enables improved predictive capability with respect to both the rst principles model and the direct modeling of the process outputs using the subspace-based approach. Table 2: Cumulative MASE Error Analysis Model Type First Principles Subspace Identication Proposed Hybrid Modeling Approach

19

Cumulative MASE (12 batches) 1620.47 72.3703 23.3738



292 8

290 288

7

Treac

286

Ca

6

5

281.8

284

281.6 685

4

400

400

705

278

4.16

300

695

280

4.2

200

282.0

282

404

500

408

600

700

800

900

276

1000

400

500

600

700

800

900

1000

Time (min)

Time (min) (a)

(b)

C ain

10 8 6 0

200

400

600

800

1000

800

1000

Time (min) 320

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 32

310 300

0

200

400

600

Time (min) (c)

Figure 4: Validation results for a fresh batch using three dierent modeling techniques: rst principles (· · · ), subspace identication (−.−) and the proposed hybrid modeling approach (−−). The process output is represented by the solid black line ( ) for comparison. Figures (a) and (b) show the prediction results (excluding the initial state estimation portion) for outputs CA (mol/m3 ) and Treac (K ) respectively. The input sequences for CA,in (mol/m3 ) (top) and Q (J/hr) (bottom) are shown in (c).

4

Applications to the PMMA polymerization reactor

In this section, the proposed methodology is applied to the PMMA batch polymerization process. To this end, rst the database is described and later the validation results for new batches are shown.

20


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4.1 Database description A rst principles model for the PMMA polymerization process is used to generate the data. This mechanistic model consists of a series of dierential equations and algebraic equations. For model building, the input-output (I/O) data was generated for 18 batches similar to the database 16 generated in an earlier study. Of these 18 batches, six batches were operated with nominal inputs perturbed by a pseudo random binary sequence (PRBS). Note that PRBS are simply used in the present work to illustrate the proposed approach. Specically designed input signals to yield rich data sets exist in the literature 39,40 . Nine historical batches were subjected to Proportional Integral (PI) set-point trajectory tracking and rest three were produced by superimposing a PRBS signal to the PI trajectory tracking input. Table 7 tabulates the composition of the training data set and also the PI settings used for trajectory tracking. Measurement noise was added to the output data. The noise is considered to be Gaussian white noise with zero mean and standard deviation reported in Table 9. As is common practice - a lter, in the present case - a moving average lter is used to attenuate the eect of the noise under PI control. All the batches were started with dierent initial conditions to imitate real life scenario. The initial batch conditions and the standard deviation data are shown in Table 8. For the testing set, I/O data was generated for six batches in a similar fashion and two batches of each type were chosen, as described earlier. The process is run for 240 minutes with data sampled every minute.

4.2 Hybrid data-driven modeling of PMMA polymerization A rst principles model is assumed to be available and, in the present example, has the same structure as the process model, but with erroneous parameter values listed in Table 10. A model with a greater number of simplifying assumptions that results in a lower value of states, or, in general, with a dierent structure can also be chosen as the rst principles model. Demonstrating the utility of the hybrid modeling approach for such a case remains the subject of future study. 21



The cross validation methodology is implemented with an initial choice of 12 Hankel rows and 3 subspace states for both the hybrid and subspace model. The 18 training batches considered in this study were divided into six groups with the random selection of three batches in each group as shown in Figure 7(a). For the rst iteration, the rst group is taken as the validation or testing set. Using the subspace identication algorithm 36 , the model is built with the remaining ve groups listed in Figure 7(b). As explained previously, the initial state estimation for the validation set is carried out with the aid of a Kalman lter. The measurement noise data used in its formulation is listed in Table 9 and process noise standard deviation is chosen to be 0.001. The initial state value of the Kalman lter is a zero vector and the error tolerance value for convergence is kept to be 0.375 0.375 0.375 . Once the outputs have converged, the model is used to predict for the rest of the batch. In cross validation methodology, as the outputs converge at dierent time instances for dierent testing groups, all the error values in the dierent simulation runs in this case are computed for the rst 100 time instances. We get the least prediction error using 8 as the number of states. For both modeling approaches (considering 8 states), the model is built (using data from all 18 batches) and the outputs are generated for the validation set (30 batches). The model predictions after convergence for a representative batch in the validation set are shown in Figure 5. The cumulative prediction error values for all the 30 validation batches for the rst principles, subspace, and hybrid approach are listed in Table 3. As can be seen from the MASE values in Table 3 and the proles in Figure 5, the hybrid model yields a signicant improvement over the subspace model alone or the rst principles models alone. Table 3: Cumulative MASE Error Analysis Model Type First Principles Subspace Identication Proposed Hybrid Modeling Approach

22

Cumulative MASE (30 batches) 1160.14 217.429 156.534


Page 22 of 32

-3.2

341.5

-3.4

341

-3.6

340.5

-3.8

log

342

340

-4

339.5

-4.2

339

-4.4

338.5 100

120

140

160

180

200

220

240

-4.6 100

120

140

160

180

200

220

240

Time (min)

Time (min)

(a)

(b)

985

340.5

980 340 975 339.5

970 965

339

960

Tjk

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Treac

Page 23 of 32

338.5

955 950

338

945 337.5

940 935 100

120

140

160

180

200

220

337 100

240

120

Time (min)

140

160

180

200

220

240

Time (min)

(c)

(d)

Figure 5: Validation results for one of the batches using the three dierent modeling techniques: rst principles (· · · ), subspace identication (−.−) and the proposed hybrid modeling approach (−−). The process output is represented by the solid black line for comparison. Figures (a), (b) and (c) show the predicted results (excluding the initial state estimation portion) for outputs Treac (K), logµ and ρ(Kg/m3 ) respectively. The input sequence for the jacket temperature Tjk (K) is shown in (d).

5

Conclusions and future work

In this manuscript, a hybrid modeling approach was presented that judiciously synergizes rst principles models with subspace-based data-driven model identication. The validity of the approach for prediction purposes (not just process monitoring) was illustrated using a CSTR example and demonstrated on a PMMA batch process example. An important aspect

23



Page 24 of 32

in the implementation of the hybrid modeling approach is that the quality of predictions is not greatly hindered by the choice of states in the residual model. Thus, in the simulation results, choosing a number of states even as low as 2 yields a model that is signicantly better than the subspace model for the process outputs. It thus provides the exibility of working with lower order models without signicantly diminishing the predictive quality. A more automated approach to select the number of states and observer parameters remains the subject of future work. In addition, the illustration of the proposed modeling approach in a feedback control scenario remains the subject of future work. Additional issues that need to be investigated include the ability to handle missing data and delays, as well as automating the choice (oine

and online) of using the data driven alone, the rst principles

alone or the hybrid model based on an appropriate monitoring framework.

6

Acknowledgments

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Engage Grant, and the McMaster Advanced Control Consortium (MACC) is gratefully acknowledged.

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model-based

2013 22 .

Appendices A

CSTR simulation Training Group

Group

Dataset Index 6 7 4 11 3 10 5 9 12 8 2 1

1

2

3

4

2

3

4 Validation Group 1

(a)

Dataset Index 11 3 10 5 9 12 8 2 1 Dataset Index 6 7 4

(b)

Figure 6: (a) A schematic showing the groups and the randomly distributed data sets used in the simulation. (b) Data set arrangement for the rst iteration in the cross validation methodology showing that the model is built using the training groups (top) and then validated on the remaining group (bottom). Training Group 2

B

Dataset Index 11 3 10 5 9 12 8 2 1 Dataset Index 6 7 4

Validation G

PMMA 3batch polymerization simulation 4 Validation Group 1

29



Page 30 of 32

Table 4: Parameter values for the CSTR example of Equation 9. Parameter F ρ V Cp ∆H k0 Tin ∆E/R

Value 50 1000 0.1 0.239 4780 7.2e9 310 5000

Unit m3 /hr kg/m3 m3 J/(kgK) J/mol hr−1 K K

Table 5: Input parameters and their bounds. Input Variable CA,in Q

Bounds [5 10 ] [300 320 ]

Unit mol/m3 J/hr

Table 6: Initial conditions of measured outputs and noise parameters. Output Variable CA Treac

Initial values , σ 2 , 0.5 310 , 5

σnoise 0.1 0.2

Unit mol/m3 K

Table 7: Model Identication database summary Input policy PI trajectory tracking( KC = −0.1, TI = 0.5) PI trajectory tracking superimposed with PRBS Nominal input superimposed with PRBS Total batches

Number of Batches 9 3 6 18

Table 8: Initial values and standard deviation data Variable Initiator concentration Monomer concentration Temperature

Value 2.06 ∗ 10−2 4.57 61

σ Units .00316 kg/m3 .07071 kg/m3 .15811 C

Table 9: Measurement noise data Output Variable Temperature Log viscosity Density

30

σ .1 .01 .1


°

Page 31 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Group 1

Table 10: The erroneous parameters used in the PMMA rst 2 values of these parameters are listed in 29 ). Parameter kd kp0 kt0 kf m kf s kθp kθt

% Mismatch 5 5 -5 5 5 5 -5

3

4

5

6

Batches 8 10 6 principles16model 3 14 4 2 18 15 12 17 5 11 13 1 7 9

Training Group

Group

Batches 8 10 6 16 3 14 4 2 18 15 12 17 5 11 13 1 7 9

1

2

3

4

5

6

2

3

4

5

6 Validation Group

(a)

1

( the original

Batches 16 3 14 4 2 18 15 12 17 5 11 13 1 7 9 Batches 8 10 6

(b)

Figure 7: (a) A schematic showing the random arrangement of batches into groups used in the simulation. (b) Arrangement of the groups for the rst iteration in the cross validation methodology showing that the model is built using the training groups (top) and then validated on the remaining group (bottom).

Training Group 2

3

Batches 16 3 14 31 4 ACS Paragon Plus Environment 2 18 15



u

yfp

u +


Page 32 of 32

yfp +

+

+

Subspace model

Observer

e=y-yfp


epred

ypred

Hybrid Modeling Approach Integrating First-Principles Models with

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