Identification for the Control of Variable Trajectories in Batch Processes

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Identification for the Control of Variable Trajectories in Batch Processes Masoud Golshan and John F. MacGregor* Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 ABSTRACT: Various issues on the closed-loop identification of empirical latent variable dynamic models for model predictive control (MPC) of the trajectories of process variables in batch systems are investigated. The concept of identifiability is explored in the context of batch processes and desirable conditions for the identification experiments to be informative for building latent variable dynamic models are proposed. It is shown that in many situations it is possible to identify the batch process models only from historical batch data without the need for external excitation of the closed-loop system. However, adding one or two batch runs with only slight set-point trajectory changes is an efficient approach to enhance the data for the identification of the batch dynamic models. The issue of model bias in closed-loop identification using nonparametric or highly parametrized modeling approaches is also investigated and it is shown that closed loop data obtained using tightly tuned PID controllers will minimize the bias.

1. INTRODUCTION In the area of process systems, the terms “identification” and “identifiable” have been used in different, but related, contexts. The first context is the identification of parameters in prespecified nonlinear models (usually theoretically based models). The second context is the identification of empirical models where the model structure and the nature of the data determine whether the empirical model can be identified. For the identification of parameters in prespecified nonlinear theoretical models, there exists a sizable amount of literature.1−6 In these papers, the identifiability of model structure is defined in terms of distinguishability of model parameters.7−9 If different values of a model parameter have distinguishable effects on the model outputs, then those parameter values are defined as distinguishable. Consequently, if all of the model parameters have the distinguishability condition, the model structure is called identifiable. In this regard, the model structure identifiability can be seen as observability with respect to model parameters.3 Most of the studies on nonlinear model structure identifiability is performed on a case by case basis for both continuous and batch processes.3,6,9−12 Batch model identification, as discussed in this paper, will refer to the second context, namely the use of experimental data to develop empirical process models for the effect of manipulated variable trajectories and disturbances on the future trajectories of certain controlled variables. Two major steps in process model identification for empirical models are the design of experiments (DOE) to collect informative data and the identification of the model structure and parameters. In the context of continuous time-invariant processes, both the process model structure and the experimental data need to satisfy a set of well-defined identifiability conditions in order to guarantee that the process model can be uniquely identified.1,2 The subject of the identifiability of empirical models from experimental data has been extensively studied since early 1970s. Most of the studies on this subject are focused on developing identifiability conditions for continuous processes. However, there are several differences between continuous and © 2012 American Chemical Society

batch processes that need to be considered when investigating the identifiability of the training data set. These include the nonlinear and time-varying dynamics of batch systems, the finite duration of each batch, and the availability of only limited numbers of batch runs to use for identification. References 13−15 discuss the effect of the information content of the training data set on the performance of system identification for linear time invariant (LTI) continuous systems and present optimal DOEs based on different objective functions, such as minimizing the bias in the identified model, generating robust models, and handling constraints. For many processes, particularly for batch processes, closed-loop identification is preferred due to safety and economic reasons. Necessary and sufficient conditions for generating informative data that satisfy the identifiability of LTI continuous systems operating in closed-loop have been derived.16−19 These identifiability conditions assume the underlying true system is LTI and the model structure is known. However, the concept of identifiability of empirical models and the design of informative experiments is not clearly defined for batch systems where a true underlying LTI model cannot be assumed. Furthermore, most concepts of identifiability of experiments are asymptotic in nature and with finite time batch processes and a finite number of batches asymptotic properties are not relevant. There is also a considerable literature on the latent variable modeling of historical batch processes for analyzing differences among batches, for monitoring of future batches, and for the prediction and control of the final product quality (e.g., refs 20−25). However, this paper is concerned with developing causal latent variable models for the set-point tracking of batch trajectories. Golshan et al.26 proposed a Model Predictive Control (MPC) approach based on Latent Variable (LV) Received: Revised: Accepted: Published: 2352

May 3, 2012 December 14, 2012 December 21, 2012 December 22, 2012 dx.doi.org/10.1021/ie301149d | Ind. Eng. Chem. Res. 2013, 52, 2352−2367

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and a true empirical model structure is assumed to be unavailable. The batch processes are also assumed to be finite time processes, and only a finite number of batch runs are available in the identification data set. Therefore, the conventional definition of identifiability conditions cannot directly apply to batch processes. However, the conventional closedloop identifiability conditions can serve this study by providing a minimal target for the conditions that the identification data set should satisfy for the batch processes. The conventional definitions of identifiability conditions for LTI systems are reviewed in this section and then used in defining desired identifiability conditions for batch processes37 (called pseudo-identifiability conditions). The empirical modeling approach considered in this study is the latent variable modeling approach,28,38−40 but many of the results should apply in general to other empirical dynamic models. 2.1. Identifiability Conditions for LTI Systems Operating in Closed-Loop. The most common definitions of closedloop identifiability and of informative experimental data is presented in Ljung et al.33 as follows. Assume the data is collected from a closed-loop system, as shown in Figure 1,

modeling of batch processes (LV-MPC) and investigated some intuitive closed-loop experiments for the identification of LV models (LVMs). Golshan et al.27,28 investigated the properties of three different latent variable modeling approaches that can be used in the course of LV-MPC. Some authors have tried to model the batch processes by Subspace Identification Methods (SIMs)29,30 that are closely related to latent variable models.31 SIMs have received considerable attention in the literature, yet the studies on SIMs are mostly confined to developing LTI models. Verhaegen and Yu32 proposed a version of SIMs for modeling Linear Time Varying (LTV) processes by fitting local models at every sample time. The approach of Golshan et al.27 models the LTV processes with much less effort, as it considers a single latent variable model for all sample times throughout the batch or batch phase, where a batch phase is any period during the total batch operating period that is identified as having with some distinctly different characteristics from other periods. In this paper, the identification of single latent variable models for all intervals within the batch or batch phase is investigated. There are no papers discussing the design of informative experiments for obtaining empirical models (including latent variable models) for batch processes from closed-loop data, nor papers on the effect of the training DOE step on the resulting model accuracy or on the controller performance. The objective of system identification in this paper is to find models that can achieve good MPC for set-point trajectory tracking and disturbance rejection in batch processes. The focus of this study is on DOE’s that satisfy the pseudo-identifiability conditions for empirical models, as defined in section 2, for batch processes in order to get as informative a data set as possible to achieve good prediction and control and to reduce the bias in the identified model. It should be noted that high prediction or control performance does not necessarily prove that the true system has been identified. However, a precise identifiability test is not possible, since this study is on modeling nonlinear and time varying batch processes using linear LV models and underlying true model parameters are unavailable. The ultimate objective of this study is to find an LV model that results in the best control using LV-MPC. Thus, our final test of any model is to evaluate it in the course of implementing the proposed LV-MPC.26 The paper is organized as follows: In section 2, the concept of identifiability is investigated and conditions to generate an informative training data set for batch process identification are proposed. In section 3, the issue of bias in closed-loop identification of batch processes is studied. Controller characteristics to be used for generating the closed-loop identification data set that reduce the bias in the identified Latent Variable Model (LVM) are proposed. The context of sections 2 and 3 are evaluated through the recently proposed LV-MPC, which is briefly reviewed in section 4 to maintain the continuity. Simulation studies are presented in section 5, and section 6 gives conclusions.

Figure 1. Schematic diagram of the closed-loop system in the training data generation step.

where G is the open-loop process, F is the controller, H is the noise dynamics, L is the set-point filter, e is the white noise, ν is the set-point of the closed-loop system, u and y are the input and output, respectively, and d is an additional dither signal that might be necessary to excite the closed-loop system. Assume the system is represented as :: y(t ) = G:(q−1)u(t ) + H:(q−1)e(t )

(1)

and the model is assumed to be 4: y(t ) = G4(q−1)u(t ) + H4(q−1)ε(t )

(2)

where : denotes System and 4 denotes Model structure. e(t) and ε(t) are white noises of different characteristics in general for the system and the model, respectively. The following definitions are necessary. Definition 1:34 DT(: ,4 ) = {θ|G4 (z) = G: (z), and H4 (z) = H: (z) a.e. z}, where θ is the vector of model parameters. The set DT consists of model parameter values that result in a model with the same process and noise characteristics as the system. Assume 0 denotes the identification method and ? denotes the identification experiment. Definition 2:34 The system : is said to be System Identifiable (SI) under 4 , 0 , and ? , SI(4 , 0 , ? ), if θ(n; : , 4 , 0 , ? ) → DT(: ,4 ) with probability 1 as n → ∞, where n is the total number of observations in the training data set. Definition 3:34 The System is said to be Strongly System Identifiable (SSI) under 0 , and ? , SSI(0 , ? ), if it is SI(4 , 0 , ? ) for all 4 such that DT(: ,4 ) is nonempty.

2. CLOSED-LOOP IDENTIFIABILITY CONDITIONS IN BATCH PROCESSES Conditions for identifying parametric models under closed-loop operation includes conditions on the model structure and on the data that will enable the process model to be identified under asymptotic conditions. This topic is widely investigated in the literature.16−18,33−36 In all of these papers, the system is assumed to be LTI. In this paper, the batch processes are considered to be nonlinear and time-varying systems in general, 2353

dx.doi.org/10.1021/ie301149d | Ind. Eng. Chem. Res. 2013, 52, 2352−2367


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Figure 2. (a) Three dimensional array of a batch process data set. (b) Observation-Wise with Time-lag Unfolded of the data set.

Definition 4:1 a signal η(k), with spectrum Sηη(ω), is Persistently Exciting (PE) of order “np” if Sηη (ω) ≠ 0 for at least “np” frequencies in the range −π < ω < π. This is a sufficient condition for identifying a model with “np” zeros and poles. However, it is preferred in practice to overspecify the order of PE. Assume that the training data set, ? , is generated by the process input that is obtained from a set of controllers switching among “r” different settings: ui = Liν − Fy i

i = 1, ..., r

that satisfy the SSI conditions for a corresponding LTI system. This is considered to be the minimum requirements for a batch process identification data set. Since a batch process is considered to be nonlinear and time varying process in general, a mathematical proof for SSI of batch systems is not possible. Therefore, one should try to satisfy as many pseudoidentifiability conditions as possible in the design of any closed-loop identification experiment on a batch process to aid in the identifiability and to further improve the quality of the identified model. If the batch process were indeed linear over its entire duration and the data set were sufficiently large (approaching asymptotic properties), then by satisfying the SSI condition the linear batch process model would be identifiable. On the other hand, if the batch process is nonlinear and/or time-varying, although one loses the formal definition of identifiability conditions, the nonlinear and time-varying behavior of the naturally occurring batch operating process actually helps in satisfying the pseudo-identifiability conditions. To better see this, consider the condition 2(a) arising from identifiability condition (4). This states that one way to guarantee SSI is to use a time-varying or nonlinear controller on a linear process. An analogous condition will exist if one has a linear controller acting on a nonlinear and/or time-varying process. This can be inferred from the fact that the final objective of the identifiability condition under closed-loop is to guarantee that the information matrix

(3)

Based on the Definitions 1−4 and eq 3, the following theorem applies.34 Theorem 1:2 for the model set defined in (2) and controller structure defined in (3) and assuming a time delay either in the system (and model) or in the controller (i.e., G(0)F(0) = 0), and the external signal (either ν or d) is Persistently Exciting (PE) of any finite order, the necessary and sufficient conditions to satisfy SSI in the Prediction Error Method (PEM) is as follows: ny + nu ≤ r(ny + nμ)

(4)

where nu and ny are dimensions of the input and output, respectively, and nμ is the summation of dimensions of setpoint and dither signal (ν + d). The proof of the above theorem is presented in ref 16. The following conclusions can be mined from Theorem 1:1,2 1. SSI cannot be guaranteed if u is determined through a noise-free linear low order pure feedback from the output (ν, d = 0). 2. Simple ways to guarantee SSI for a SISO system: (a) Use a controller that shifts between different settings during the generation of identification experiments (or equivalently use a nonlinear, or time varying controller). (b) Add a PE external signal (dither signal, d, or timevarying set-point signal, ν). 2.2. Pseudo-identifiability Conditions for Batch Processes Operating in Closed-Loop. In this study we refer to “pseudo-identifiability conditions” for batch processes as those

⎛ −y 2 (t ) −y(t )u(t )⎞ ⎟ E⎜⎜ ⎟ u 2 (t ) ⎠ ⎝−u(t )y(t )

is not singular.2 In order to achieve this objective, one should ensure that y(t) and u(t) are not linearly dependent. References 1 and 2 assume that the system is LTI and show that switching between r different linear controllers (eq 3) or having a nonlinear controller satisfies this requirement. By analogy, a nonlinear process generating y together with a linear controller for u will satisfy the same condition. Hence, it can be argued that the normal closed-loop batch operating data using a linear controller with no external excitation satisfies the conditions for 2354



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LTI model identification under closed-loop (i.e., satisfies the pseudo-identifiability conditions). Note that all modeling approaches including the latent variable methods presented in this study are deviation models where the mean or set-point trajectories are removed from the batch data set before performing the LV modeling. However, the time-varying set-point trajectories cause the operating conditions of the batch to move over a range of conditions, thereby making the deviation form of the model more nonlinear and time-varying, again helping to satisfy the pseudoidentifiabilty condition. The most common approach to satisfy the SSI conditions for data sets obtained from closed-loop system is to add an external excitation signal on top of the controller output (d) or into the set-point (ν) (condition 2(b) in section 2.1). For batch processes, an easy approach is to run one or more batches with slightly different set-point trajectories. The necessary conditions on these set-point trajectories will depend upon the structure of the latent variable model, as shown later. Although, as discussed, batch processes are often identifiable from historical data without the need for this additional excitation of the closedloop system by an external signal, these additional excitations should improve the quality of the model estimation. In the rest of this section, experiments that satisfy the SSI conditions (pseudo-identifiability conditions for batch processes) for three LV modeling alternatives used in LV-MPC27,28 are proposed as different outcomes from the Theorem 1. 2.2.1. Observation-Wise with Time-lag Unfolding (OWTU). The data set of a batch process is naturally represented as a three-dimensional array, since there are three dimensions in the batch data set, as shown in Figure 2a. In order to apply a LVM such as Principal Component Analysis (PCA) that is proposed to be used in the course of the LV-MPC, one needs to unfold the three-dimensional data set into a two-dimensional matrix. Different unfolding approaches lead to different modeling properties. OWTU, illustrated in Figure 2, is one of the LV modeling approaches studied in refs 27 and 28. Parameters “ph” and “f h” are past and future lags respectively considered in the OWTU approach. This unfolding approach is a similar approach, in terms of data arrangement and modeling properties, to conventional time-series analysis and system identification studies. It leads to a LTI dynamic model of the batch process over each phase and captures the average dynamics of the batch over the considered phase. However, the main difference between LV modeling on OWTU data set and conventional time-series analysis is that LVMs try to find a model for the complete batch (phase) considering all variables together in the multivariate model format. To ensure identifiability conditions for a PCA model built from OWTU, one needs to satisfy either of conditions 2(a) or 2(b) arising out of Theorem 1 in section 2.1. Outcome 1 therefore defines the SSI conditions for the data set resulting from this OWTU modeling approach. It should be noted that implementation of any control algorithm, including the LV-MPC,26 on batch processes can be performed in multiphase fashion, which means the completion time of a batch process can be split up into many phases and a batch in each phase is considered as a separate batch process, and a separate modeling and control is performed on each phase. The concept of multiphase modeling and control of batch processes is discussed in refs 26 and 41. Outcome 1: If the system is square (ny = nu = nν), the pseudo-identifiability condition is satisfied if at least one of the following conditions is satisfied:

(a) The System is time-varying and/or nonlinear over the duration of each batch phase or within the phase from batch to batch. (b) The set-point is different at more than “1 + ph + fh” sample times throughout the batch phases or within the phase from batch to batch. In this study, the system is assumed to be square (the number of manipulated variables and controlled variables are assumed to be equal, and there is a set-point for every output). Thus, either Outcome 1(a) or Outcome 1(b) should result in an informative identification data set. However, considering the fact that both of these conditions are mild, satisfying both of conditions presented in the Outcome 1 will further improve the richness of the identification data set. As long as the set-point (and hence operating region) is time-varying, the nonlinear and time-varying effects are included in the data set. Thus, the pseudoidentifiability condition is satisfied according to condition 2(a) of Theorem 1. However, inclusion of extra set-point trajectories that are different from the main set-point trajectory at several sample times during each modeling phase will further improve the quality of the identified model or provides the SSI conditions in case of having a LTI process. Note that the latent variable methods model the data set by considering all columns of the matrix and extracting fewer LVs to explain the major variations in the matrix. The maximum number of LVs is equal to the number of columns. Different (time-varying) set-points at more than “1 + ph + f h” sample times ensures that the columns of the OWTU matrix are linearly independent (OWTU is a full rank matrix). However, in order to better estimate the model parameters, it would be desirable to have the set-point trajectory different at significantly more than “1 + ph + f h” sample times. Another alternative to ensure identifiabilty for a LTI process is to satisfy condition 2(a) of Theorem 1namely, switch between different controller settings during each phase (or have a time varying controller). However, as pointed out earlier, this identifiability condition would equivalently be satisfied if the batch process is a nonlinear or time-varying process and the controller and model are LTI, as it is with a PCA model on the OWTU data. In practice, it would generally be prudent to rely not only on one of these identifiability conditions, but to satisfy both by not only relying upon the nonlinear time-varying behavior of the batch process, but also by introducing one or more set-point deviations from the nominal set-point trajectories within each batch phase or from batch (phase) to batch (phase). 2.2.2. Batch-Wise Unfolding (BWU) and Regularized Batch-Wise Unfolding (RBWU). BWU and RBWU are two of the three modeling candidates studied in refs 27 and 28. These unfolding approaches are shown in Figures 3 and 4, respectively. The parameter “L” is the number of repetition of each batch in the RBWU, as explained in refs 27 and 37 to introduce the regularization to the batch data set. It can be thought of in two ways: One can start with the BWU shown in Figure 3b and then replicate each row L times while shifting it by one interval in each case. Alternatively, one can start with the OWTU shown in Figure 2 and use the past and future horizons (ph, f h) to cover the (K−L) time steps of the batch in each row and use only L block rows. Since these two modeling approaches (BWU and RBWU) share most of the same properties, they are studied together. For a detailed comparison of BWU, RBWU, and OWTU, the reader is referred to refs 27 2355



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Figure 3. (a) Three dimensional array of a batch process data set. (b) Batch-Wise Unfolding of a batch process data set (c) matrix of LV scores.

Figure 4. (a) Three dimensional array of a batch process data set. (b) Regularized Batch-Wise Unfolding of a batch process data set. (c) Matrix of LV scores.

and 42. RBWU and OWTU are variants of Batch Dynamic Model in Camacho et al.42 As explained in refs 27 and 37, using the BWU or RBWU modeling approaches, the PCA method models the timevarying and nonlinear properties of a batch process by modeling all local variations at every sample time. As a result, the concept of identifiability condition applies only locally to the BWU and RBWU modeling approaches. This means that the SSI condition should be satisfied at every sample time for different observations (from batch to batch). Thus, assessing Theorem 1, the DOE to satisfy the SSI condition for BWU and RBWU data sets is as follows: Outcome 2: “If the system is square (ny = nu = nν), the SSI condition is satisfied if the set-point signal is time-varying at every sample time from batch to batch in the training data set.” In the LV modeling by BWU and RBWU approaches, the process is modeled locally at every sample time throughout the batch (phase). However, it is different from local modeling of processes by conventional modeling approaches where every local model is developed independent of other local models and thus nonlinear behavior of the batch process does not help for modeling at every sample time. Multivariate latent variable modeling approaches consider variables at all sample times in one model by summarizing the fat matrices in Figures 3b and 4b by smaller matrices of LV scores (T) (Figures 3c and 4c). Therefore, the nonlinear properties still help in satisfying the identifiability conditions. However, this effect cannot be quantified theoretically at this time. Thus, in order to ensure the pseudo-identifiability conditions, at least one different setpoint should be considered in the data set in the way that the different set-point has a deviation from the original set-point at all sample times so that each column in the unfolded array contains at least one set-point change (PE of order 1 at each time step). If not (i.e., if the different set point trajectory

overlaps the nominal one over a certain period), then one might lose identifiability during that period. This condition can be achieved by running a batch using a shifted nominal set-point trajectory. However, such a setpoint trajectory may result in a slightly different final product quality that may not be desirable for industrial applications. An alternative approach would be to use set-point trajectories from two different product grades or ask an expert to help in selecting a modified set-point trajectory in a way that the final product quality should stay in the specified range.

3. DESIGN OF EXPERIMENTS (DOE) TO REDUCE THE BIAS IN THE CLOSED-LOOP IDENTIFICATION BY LVMS Closed-loop identification can often result in bias in the identified model. For parametric methods such as PEM, identifiability conditions such as discussed in section 2.1 define the necessary and sufficient conditions for asymptotic identifiability (and hence lack of bias for large data sets) of LTI systems. However, for finite data sets, as available in batch phases, one can still see bias even if the LTI identifiability conditions are satisfied. Furthermore, if the model structure is not defined correctly, the model will be a biased one. Therefore, with fitting linear models to time varying and nonlinear systems some model bias from the true system is inevitable. This problem is extensively studied for parametric identification methods.1,2,43,44 With nonparametric modeling approaches such as Correlation Analysis Methods (CAM) and Spectral Analysis Methods (SAM), one avoids the problem of model structure definition, but even for LTI systems, the method inherently includes bias under closed-loop identification.1,2,34 Such a bias can also be expected for highly parametrized models such as ARX and latent variable models. For a closed loop system, Figure 1, 2356



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(assuming L(q−1) = 1 without loss of generality) the following equations can be written: y(t ) =

⎤ ⎡ 1 1 −1 ⎥, ⎢ G ( q ) ν ( t ) + z ( t ) s 1 + Gs(q−1)F(q−1) ⎣ F(q−1) ⎦

z(t ) = F(q−1)HS(q−1)e(t ) u(t ) =

expression for nonlinear time-varying systems, they can provide practical justifications and guidelines for DOE in batch process modeling using LVMs. From these equations, one can see that if the ratio Φz/Φν (disturbance to signal ratio) is small, the bias term is small and a close approximation of the open-loop transfer function is obtained using nonparametric methods.45 Therefore, as long as the noise level (e) is small compared to the external excitation (ν) at any frequency, there will be small bias at those frequencies. Therefore, the inclusion of different set-point trajectories in the training data set for a batch process will decrease the disturbance to signal ratio (Φz/Φν) ratio leading to a smaller bias. The controller and disturbance dynamics also will have an important effect on the bias in the identified model. To better see this, eq 11 can be rewritten as

(5)

1 [ν(t ) − z(t )] 1 + Gs(q−1)F(q−1)

(6)

Hence, it can be shown that the spectra of the input Φu and cross-spectra between the input and output Φyu are given by 1 [Φν(ω) + Φz (ω)] |1 + Gs(eiω)F(eiω)|2

Φu =

(7)

⎡ ⎤ 1 1 G (e−iω)Φν − −iω Φz ⎥ iω iω 2 ⎢ s F (e ) ⎦ |1 + Gs(e )F(e )| ⎣

Φyu =

Ĝ (e−iω) = (8)

γu =

(1 + Gs(q−1)F(q−1))2

γyu =

[γν + γz]

(9)

⎡ ⎤ 1 ⎢Gs(q−1)γν − γ⎥ −1 z (1 + Gs(q )F(q )) ⎣ F (q ) ⎦ −1

2

(10)

As a result, the estimated transfer functions from the spectral analysis approach and the correlation analysis approach respectively are obtained by eqs 11 and 12:2 Ĝ (e

−iω

)=

h(̂ q ) = −1

Φyu Φu

ru−1ryu

=

=

1 Φ F(e−iω) z

Gs(e−iω)Φν −

Φν(ω) + Φz (ω)

Gs(q−1)γν −

(11)

1 γ F(q−1) z

γν + γz

Gs(e−iω)Φν − F(eiω)H(e−iω)ΦeHT(eiω) Φν(ω) + Φz (ω)

Equation 13 shows that a high magnitude of the disturbance (H(eiω)) and of the controller (F(eiω)) at certain frequencies will increase the bias at those frequencies. The nature of the disturbance in most batch processes is not clear due to the time-varying and nonlinear behavior of the process and generally poor knowledge of how the disturbances enter and propagate. The effect of disturbance and controller characteristics on the identification studies used in this paper can be explained in general, as follows. 3.1. Effect of Disturbances (H(eiω)) on Possible Model Bias. The latent variable models used in the LV-MPC methodology are expressed in terms of deviations of the output variables (y’s) from their set-point trajectories and in terms of the deviations of other variables from their mean trajectories.26,27 Under these conditions, any deviation of the output variables from their set point trajectories that cannot be explained by the input (u) trajectories is absorbed into the disturbance term (H(q−1)e). The disturbance term would include real disturbances arising from the propagation of different initial conditions (recipe, raw material, and impurity variations) for each batch and from disturbances entering during the progress of the batch (e.g., impurities in a feed stream entering a semibatch process). However, under closedloop identification with mean centering “y” by “ysp”, the disturbance term (H(q−1)e) would also have to absorb any offset in the y’s from their set-point trajectories that arises from the inability of the existing controller to track the set-points during the runs used for closed-loop identification (a pseudodisturbance). This latter contribution can be quite large in the case where a PI controller is being used to track ramp set-point trajectories since a PI controller cannot keep up with ramp setpoints unless the process has an integrator, which is uncommon in chemical processes. As a result, this deviation will appear as a nonstationary disturbance and will inflate H(eiω), particularly at low frequencies. According to eq 13, this would lead to an apparent model bias in the model gain. 3.2. Effect of the Controller Tuning (F(eiω)) on Possible Model Bias. The nature and tuning of the controller used during the closed-loop identification studies will also affect the model bias. A tightly tuned PI controller (large PI gains) will result in a large magnitude of the controller frequency response (term F(eiω) in eq 13), hence, increasing the magnitude of bias.

1

−1

Φu

=

(13)

and the auto and cross-correlation functions are obtained for “u” and “yu” are as follows: 1

Φyu

(12)

Clearly, the estimated frequency response (Ĝ ) is a biased estimate of the true response (Gs) and ĥ(q−1) is a biased estimate of Gs(q−1). The direct consequence of the above discussions is that the identified latent variable models will exhibit some bias under the proposed closed-loop identification approach. Latent variable models are nonparsimonious models that are expected to exhibit some characteristics of nonparametric models and their bias relationships shown in eqs 11 and 12. For instance, the LVM obtained from PLS is the generalized version of the ARX model found by eq 12, in which the output is expressed as a function of transformed version (through principal components) of the past inputs, output, and possible extra measured variables. In a PCA model of the batch process, the same concept applies with the modification of modeling the effect of future variables as well as past variables. They are also based on finite data sets from the batch phases, and the true process is time-varying and nonlinear. The objective of this section and the simulation section to follow is to investigate this bias and to identify conditions under which the bias can be minimized. Equations 11 and 12 are valid for nonparametric models of LTI systems. Although there are no equivalent analytical 2357



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Figure 5. LV-MPC based on BWU modeling approach, using 40 historical batches (with PI controller) plus one extra batch run (with PI controller) on a similar but lower set-point trajectory throughout the batch.

Figure 6. LV-MPC based on BWU modeling approach, using 40 historical batches plus one extra batch run with a higher set-point trajectory early in the batch and a lower set-point trajectory for the remainder of the batch.

However, a tightly tuned PI controller will reduce the magnitude of the pseudo-disturbance term (H(eiω)e) discussed by decreasing the offset in the y’s from their corresponding set-points. The latter effect has a stronger impact on the bias than the magnitude of F(eiω), as, according to eq 13, the square factor of the H(eiω) appears in the bias term. Thus, a tightly tuned PI controller should be preferred over a sluggishly tuned PI controller in general because it reduces the magnitude of the low frequency components in the pseudo-disturbance term as discussed. This would imply that the better the controller used during the identification experiments, the better the identified model will be and hence the better the LV-MPC designed from it will be. This raises a problem because, the better the existing controller, the less need for a LV-MPC. In practice, this probably means that one might best iterate on the model building, first using batch data collected from the existing controller, then once the improved LVMPC is designed and implemented, collect more closed-loop data to add to the earlier data and then reidentify the model. This will

also enhance the identifiability conditions described in section 2 by using more than one controller for collecting identification data.

4. REVIEW OF THE LV-MPC METHODOLOGY The LV-MPC methodology proposed in ref 26 is used in this study to evaluate the theoretical aspects presented in sections 2 and 3. It is briefly reviewed as follows in order to maintain the continuity. Interested readers should refer to refs 26 and 27 for details of the LV-MPC algorithm and many more examples of the application of the LV-MPC. Consider the PCA model developed based on a batch data set (BWU, OWTU, or RBWU), where the objective is to track specified set-point trajectories in a new batch run subject to possible disturbances.26 ζk (as defined by eq 14) contains all the information available at sample time k. T T T T ζkT = [xme , k , ycv , k , uc , k , ysp , k ]

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Figure 7. (a) Performance of the PI controller on the similar but lower set-point trajectory. (b) Performance of the PI controller on the set-point, which is above the original set-point for the first half of the batch and then smoothly switches to below the original set-point trajectory for the rest of the batch.

where xme, ycv, uc, and ysp are vectors of measured, controlled, manipulated, and set-point variables, respectively. Assume the control algorithm at sample time k of a batch phase. The complete batch phase data contains the measurement vectors ζ1 to ζK and can be separated as follows:

where xP1 and xP2 are vectors of known (past) information and xf1 and xf2 are vectors of unknown (future) information at time k: xTp1 = (ζTj |j=1:k−1, xTme,k, yTcv,k, yTsp,k), xTp2 = (yTsp,j|j=k+1,...,k+K), xTf1 = T T T (uc,k , uc,jT |j=k+1,...,k+K−1, xme,j |j=k+1,...,k+K), xfT2 = (ycv,j |j=k+1,...,k+K). The corresponding loadings, P matrix in the PCA model can also be separated in the same way as the “x” vector as follows:

xT = [ζ1T , ζ2T , ..., ζkT , ..., ζKT ] =

[ζjT | j = 1: k − 1 ,

Pk = [PP1, k ; PP 2, k ; Pf 1, k ; Pf 2, k]

T T T T T xme , k , ycv , k , ysp , k ysp , j| j = k + 1,..., k + K uc , k ,

It is shown in refs 26 and 37 that the process input and output can be written as functions of the latent variable scores in the PCA model as follows:

T T × ucT, j| j = k + 1,..., k + K − 1 , xme , j| j = k + 1,..., k + K ycv , j | j = k + 1,..., k + K ]

= [xPT1 , xPT2 ; xTf 1 , xTf 2]k

(16)

xf̂ 2 = Pf 2(PTf Pf )−1(τk̂ + Δτk̂ − PpTxp , k)

(15) 2359

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Figure 8. LV-MPC based on BWU modeling approach, using 40 batches with plus one extra batch run with the lower set-point trajectory. An RBS dither signal (d) signal was also delivered on top of the controller output for all batch runs.

performance index

Figure 5

Figure 6

Figure 8

NMPC

1 1 (ycv̂ − ysp)T V1(ycv̂ − ysp) + uTf̂ V2uf̂ Δτk̂ 2 2 1 1 = (xf̂ 2 − xP 2)T V1(xf̂ 2 − xP 2) + uTf̂ V2uf̂ 2 2

RMSE of (y − ysp) STD of Δu

0.6944 6.5792

0.7041 6.8985

0.6925 6.6046

0.1168 4.5246

xPT = [xPT1 xPT2]

Table 1. Comparison of the LV-MPC Performance Based on Different Alternative Identification Data Set with the Ideal NMPC

uf̂ = Puf (τk̂ + Δτk̂ )

min

(19)

xTf = [xTf 1 xTf 2] (18)

PpT = [PPT1 PPT2]

The past data can be used to estimate the score vector of the current batch, τ̂k, using a missing data imputation method, which summarizes the current position of the batch.26,46 Once the current score of the batch is estimated, a correction (Δτ̂k) can be made to the current score of the batch to bring it to the desired value. This can be achieved by formulating an optimization problem where the objective function can be represented as follows:

PTf = [PTf 1 PTf 2]

Combining eqs 17, 18, and 19 and solving the optimization problem analytically, one can obtain the optimum Δτ̂k. Δτ̂k contains information necessary to compute adjustments (eq 18) to all future inputs until the end of the batch phase (“infinite” horizon control26). Note that if there are constraints on uf (eq 18)

Figure 9. LV-MPC based on RBWU modeling approach, using 20 similar batches plus one similar but shifted set-point trajectory, regularizing timelag (L) = 5, STD of Δu = 6.5075, RMSE of (y − ysp) = 0.6807. 2360



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Figure 10. LV-MPC based on OWTU modeling approach, using two historical batches in the training data set run with PI controller, STD of Δu = 6.4766, RMSE of (y − ysp) = 0.9938.

set-point trajectory being unnecessary but helpful, as explained in section 2. Figures 5 and 6 show the performance of the LV-MPC methodology based on the BWU modeling approach using 40 historical batches plus one batch run with a slightly different set-point trajectory. Two alternatives for considering the different set-point trajectory are tested. In Figure 5, the one different set-point in the training data set is similar to the original set-point but shifted slightly lower, while in Figure 6 the different set-point is similar to the original set-point trajectory but shifted higher during first half of the batch and lower for the rest of the batch (a possible example of changing the set-point while trying not to change the final product quality). Figure 7 shows the shape of the different set-point trajectories studied in Figures 5 and 6 as well as the performance of the PI controller on the two different set-point alternatives. Figure 8 is the same as Figure 5, but an RBS dither signal (d) is also delivered on top of the controller output in the training data generation step for all batches to excite the closed-loop system. As discussed in section 2, this is a common method to satisfy the identifiability conditions. Table 1 compares the performances of Figures 5, 6, and 8 to a noise free Nonlinear MPC (NMPC) based on a perfect mechanistic model (best achievable feedback control). It shows that the performance of all above alternatives (Figures 5, 6, and 8) are in the same range. Specifically, Table 1 shows that use of the RBS dither signal (d) during identification did not improve the performance of the resulting LV-MPC. This is an interesting result since most industries do not like the use of an external dither signal in the data generation step since it introduces constant variation and the resultant batch may result in off-spec product. This study shows that one can get similar results using historical batches with a modest requirement of having one slightly different set-point trajectory for the same product or using two different product grades with slightly different setpoint trajectories. Figure 9 shows the performance of the LV-MPC based on RBWU modeling approach. It is seen that this approach leads to a similar model quality (even slightly better) than the BWU

the optimization problem must be solved numerically. Once Δτ̂k is calculated, the corresponding ûf can be computed using the PCA model (eq 18) and its first element can be implemented on the process. At the next sample time, the same procedure can be repeated till the end of the batch (phase). In the control formulation, the matrices V1 and V2 should be chosen carefully. A good choice could be to pick V1 as a diagonal matrix with the subsequent diagonal entries reducing exponentially to put more penalties on the early future values rather than distant values. The matrix, V2, however, should be chosen to appropriately penalize the rate of changes in the MV’s.26

5. SIMULATION STUDIES To illustrate the theoretical and intuitive points discussed in sections 2 and 3, simulation studies are conducted. The case study is the temperature control problem in a batch reactor studied in refs 26, 47, and 48, in which the temperature of a batch reactor is controlled by manipulating the jacket temperature set point. In section 5.1, simulation studies are presented to confirm the identifiability of batch processes using only historical batches in the training data set as discussed in section 2 and the impact of incorporating into the training data a batch run with a different set-point trajectory. Section 5.2 addresses the discussions of section 3 by focusing on the PI controller tuning in the identification data generation step. Three simulation examples are used to show the performance of the LV-MPC algorithm based on three different PI tunings. 5.1. Identifiability Tests. Golshan et al. 27,28 used simulation studies to show that batch processes can be identified only from historical batches without the need for external RBS excitation (d or ν in Figure 1). This observation motivated the studies of this paper. Theoretical and intuitive arguments presented in section 2 implied that not only is there no need to use an RBS dither excitation but also inclusion of only one batch with a different set-point trajectory is often enough to get a reasonable model based on BWU and RBWU. For the OWTU modeling approach, identification should be possible only using historical data, the inclusion of a different 2361



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Figure 11. Trajectory tracking by (a) loose PI controller used in the identification data generation step, (Kc = 10, τi = 30). (b) LV-MPC based on a PCA model obtained from batch-wise unfolded closed-loop data which is generated by the loose PI controller.

model of the batch process for the purpose of prediction. Figures 11 to 13 show the effect of the PI tuning used during the training data generation step on the performance of the resultant LV-MPC. Table 2 summarizes performances of the LV-MPC and PI for trajectory tracking corresponding to Figures 11−13. The above simulations show that the most tightly tuned PI controller in the identification data set with large Kc and small Ki leads to the best trajectory tracking performance by the LVMPC and the loose PI tuning leads to a biased model that results in an obvious bias in the trajectory tracking when the model is used in the course of LV-MPC. Flores-Cerrillo and MacGregor49 also claimed that a fast PI controller in the identification experiment leads to a better trajectory tracking by the LV-MPC as compared to a sluggish controller in the previous version of LV-MPC methodology without further explanation of this observation. An explanation for the

with the same DOE requirements but with a fewer number of batch runs in the training data set. Figure 10 shows the performance of the LV-MPC based on OWTU using only two historical batches with no set-point trajectory change in the training data set. 5.2. Effect of Controller Characteristics. In this section, the rationalization presented in section 3 about the effect of controller and disturbance dynamics on the magnitude of bias in the identified model will be tested by using different PI controller tunings in the training data generation step. The LVMPC algorithm based on the BWU modeling approach using 40 batches run on similar set-point trajectories plus one batch run with slightly different set-point trajectory (as in Figure 5) will be used to test the performance when different PI controllers are used in the identification. The noise level on top of the controlled variable (Tr) is considered to be small (noise level ≈ 0.1−0.2 °C). The LV-MPC methodology utilizes a PCA 2362



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Figure 12. Trajectory tracking by (a) tight PI controller used in the identification data generation step with moderate Kc and larger Ki (Kc = 40, τi = 4). (b) LV-MPC based on a PCA model obtained from a model using batch-wise unfolded closed-loop data generated by the tight PI controller.

results in a lower controller gain in the low frequency term in the Bode diagram and according to eq 13 this should lower the bias term (second numerator term) in this frequency range. The frequency response of the transfer function of the batch process under study can be crudely approximated by a first order process with its dominant time constant is also shown in Figure 14. Figure 15 shows the pseudo-disturbance dynamics explained in section 3 resulting from the different PI controllers used for tracking the set-point ramps in Figure 11 to 13 as well as the spectrum of the pseudo-disturbance for the three PI examples. In the process operating region, which is low to moderate frequencies, the slow PI has the smallest magnitude in the Bode plot. However, according to Figures 11a and 15a, the slow PI operating on a set of ramps results in a large persistent offset in the trajectory tracking that the PI is unable to remove it. As explained in section 3, during identification this persistent offset

improved results from using a tightly tuned controller was provided in section 3. These simulation studies provide an illustration of this effect. Recently, Bakke et al.50 also investigated the effect of different PI tunings in the closed-loop training data generation on the identified model for continuous processes where the external exciting signal is added on top of the setpoint variable and also concluded that a large controller gain leads to a more accurate model. In order to better explain the results of Figures 11−13, the frequency responses of different PI controllers used in these figures are illustrated by plotting their Bode gain diagrams in Figure 14. The large gain in the low frequency ranges of Figure 14 represents the effect of the integral term. If the integral gain is large, the magnitude of the controller frequency response is large in the low to moderate frequency ranges. The horizontal lines correspond to the proportional gain of the PI controllers, which is an all frequency pass filter. Thus, a small integral gain 2363



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Figure 13. Trajectory tracking by (a) tight PI controller used in the identification data generation step with larger Kc and smaller Ki (Kc = 60, τi = 120). (b) LV-MPC based on a PCA model obtained from batch-wise unfolded closed-loop data generated by this tight PI controller.

appears in the bias term (see eq 13), it dominates the effect of “F” and leads to a large bias. Thus, a tightly tuned PI controller is preferred over a loosely tuned PI controller for batch model identification. Comparison of the two tightly tuned PI controllers, according to Figures 14 and 15, shows that the tightly tuned controller with larger Ki and smaller Kc results in a larger magnitude in the low frequency region for both the controller frequency response and the corresponding pseudo disturbance. Thus, a tightly tuned PI controller with larger proportional gain and smaller integral gain results in the smallest bias in the identified model. If one can use accurate feed-forward information for trajectory tracking, it is possible to decrease the gain of the feed-back controller while avoiding the large offset in the trajectory tracking. In this case, the bias would become smaller as both magnitudes of “F ” and “H” get smaller in the frequency space.

Table 2. Summary of Performance of the LV-MPC Based on a PCA Model Built on Closed-Loop Data Generated by Different Tunings of a PI Controller loose PI in the data tight PI with moderate tight PI with large Kc generation Kc and larger Ki in the and smaller Ki in the step data generation data generation RMSE of (y − ysp) from PI RMSE of y − ysp) from LV-MPC

2.9875

1.2251

1.0512

2.4820

0.8897

0.6944

will have to be absorbed into the disturbance term (H(eiω)e) when y is mean centered by ysp in the modeling step (pseudodisturbance). Figure 15b shows that the magnitude of the pseudo disturbance resulting from the loose PI controller is significantly larger than that resulting from tight PI controllers in the low frequencies. As the square factor of the pseudo-disturbance 2364



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Figure 14. Bode diagram for different PI controllers used in the data generation step and an approximated process transfer function.

Figure 15. (a) Pseudo-disturbance resulting from mean centering y with ysp. (b) Spectrum of the pseudo-disturbance.

the general identifiability theorems for LTI systems. Under such conditions, the training data set is informative enough for finding an adequate linear model for the batch process to be used in the course of the LV-MPC. It is shown that most batch processes are identifiable from only the historical data set and there is no need for addition of external RBS dither signal to the closed-loop system during the training data generation step. The maximum requirement would be to have an identification data set that includes data from more than one set-point trajectory in the training data set, a modest requirement. The bias issue in closed-loop identification using LVMs is also studied. The controller characteristic to be used in the training data generation step that will lead to small bias in the identified model is explored. It is shown that if the PI controller is used to generate the identification data set, a tightly tuned PI controller with high proportional gain and low integral gain will lead to latent variable models having less bias than the LVMs resulting from other PI tunings. Therefore, better LV-MPC performance for trajectory tracking is achieved.

It is also seen in Figures 11 to 13 that the LV-MPC produces a better performance for the trajectory tacking as compared to the PI controllers. Thus, one should use the best possible PI controller according to the above guidelines for generating the identification data set to build the LV model and also reidentify the PCA model after collecting more closed-loop data from the improved control by the LV-MPC.

6. CONCLUSION Closed-loop identification of batch processes has to be carried out with special considerations. The identification experiments have a strong impact on the ability to identify empirical models and on the quality of the identified model. It is shown that although the conventional definition of identifiability conditions for LTI systems does not apply to batch processes, it provides a set of desirable conditions that will help to ensure the identification of batch models. A set of “pseudoidentifiability conditions” that satisfies the strong system identifiability conditions for different latent variable model alternatives studied in this paper is proposed as outcomes from 2365



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Identification for the Control of Variable Trajectories in Batch Processes

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