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Real-time feasible multi-objective optimization based nonlinear model predictive control of particle size and shape in a batch crystallization process
Abstract
This paper presents nonlinear model predictive control (NMPC) and nonlinear moving horizon estimation (MHE) formulations for controlling the crystal size and shape distribution in a batch crystallization process. MHE is used to estimate unknown states and parameters prior to solving the NMPC problem. Combining these two formulations for a batch process, we obtain an expanding horizon estimation problem and a shrinking horizon model predictive control problem. The batch process has been modeled as a system of differential algebraic equations (DAEs) derived using the population balance model (PBM) and the method of moments. Therefore, the MHE and NMPC formulations lead to DAE-constrained optimization problems that are solved by discretizing the system using Radau collocation on finite elements and optimizing the resulting algebraic nonlinear problem using Ipopt. The performance of the NMPC–MHE approach is analyzed in terms of setpoint change, system noise, and model/plant mismatch, and it is shown to provide better setpoint tracking than an open-loop optimal control strategy. Furthermore, the combined solution time for the MHE and the NMPC formulations is well within the sampling interval, allowing for real world application of the control strategy.
... However, for many processes, it is not possible to accurately measure all states on-line, and model parameters may change from batch to batch. This challenge drives the need for a state estimator (Cao et al., 2017). The Extended Kalman Filter (EKF) (Prasad et al., 2002) and Luenberger observers (Motz et al., 2008) are popular state estimator for unconstrained systems. ...
... The Extended Kalman Filter (EKF) (Prasad et al., 2002) and Luenberger observers (Motz et al., 2008) are popular state estimator for unconstrained systems. However, because of the highly non-linear dynamics and hard constraints, non-linear Moving Horizon Estimation (MHE) is more appropriate than EKF in the batch crystallization model (Cao et al., 2017). The MHE is an optimization based method including the nonlinear process model, which uses measurements gathered in a certain time interval for the observer correction (Mesbah et al., 2011;Szilágyi et al., 2018). ...
... In this method, the whole input profile in the interval t k ; t f  à is computed, but only the control action in the interval t k ; t kþ1 ½ Þis implemented. At the next sampling instance t kþ1 , the control interval moves from t k ; t f  à to t kþ1 ; t f  à and the optimal control problem is reevaluated with new state estimates to update the whole input profile in the interval t k þ 1 ; t f  à (Cao et al., 2017). ...
Crystallization is a complex process of heat and mass transfer, its product quality is affected by many factors, thus it is a challenge to control this process. Modelling and optimization of the manipulated variables is a feasible way to improve the product quality of crystallization. In this paper, a modified numerical scheme is developed to solve the process model of anti-solvent batch crystallization. The proposed numerical scheme can handle growth, nucleation, agglomeration and breakage kinetics by means of the Method of Characteristics (MOC), which provides an accurate and efficient result of the crystal size distribution (CSD). Then Non-linear Model Predictive Control (NMPC) and non-linear Moving Horizon Estimation (MHE) are presented to provide a temperature profile for optimizing the CSD. The result of case studies indicates that the calculated results of the MOC modified process model were in agreement with the experimental results, which indicated that the MOC method could deal with the specific crystallization process efficiently and accurately. The NMPC-MHE optimized temperature control strategy was tested using the verification experiment of anti-solvent crystallization of β-Artemether, the validated result indicated that the method developed had high feasibility and accuracy. The computational time for the MHE and NMPC formulations with 90 control and sampling steps is well within the sampling interval, which allows for real time process control.
... où les expressions des modificateurs sont données par : nécessite la connaissance des gradients de la fonction objectif et des contraintes du procédé pour chaque itération de la méthode (Cao et al., 2017). Ces gradients peuvent être obtenus par des méthodes d'estimation du gradient ou encore à partir des mesures des sorties et de l'estimation des gradients des y p . ...
... Les résultats du problème d'optimisation peuvent être envoyés directement à un système de commande (Cao et al., 2017). La différence entre la D-RTO et la RTO classique est que la D-RTO traite des systèmes dynamiques (Biegler, 2014). ...
D'une manière schématique, l'optimisation dynamique de procédés consiste en trois étapes de base : (i) la modélisation, dans laquelle un modèle (phénoménologique) du procédé est construit, (ii) la formulation du problème, dans laquelle le critère de performance, les contraintes et les variables de décision sont définis, (iii) et la résolution, dans laquelle les profils optimaux des variables de décision sont déterminés. Il est important de souligner que ces profils optimaux garantissent l'optimalité pour le modèle mathématique utilisé. Lorsqu'ils sont appliqués au procédé, ces profils ne sont optimaux que lorsque le modèle décrit parfaitement le comportement du procédé, ce qui est très rarement le cas dans la pratique. En effet, les incertitudes sur les paramètres du modèle, les perturbations du procédé, et les erreurs structurelles du modèle font que les profils optimaux des variables de décision basés sur le modèle ne seront probablement pas optimaux pour le procédé. L'application de ces profils au procédé conduit généralement à la violation de certaines contraintes et/ou à des performances sous-optimales. Pour faire face à ces problèmes, l'optimisation dynamique en temps-réel constitue une approche tout à fait intéressante. L'idée générale de cette approche est d'utiliser les mesures expérimentales associées au modèle du procédé pour améliorer les profils des variables de décision de sorte que les conditions d'optimalité soient vérifiées sur le procédé (maximisation des performances et satisfaction des contraintes). En effet, pour un problème d'optimisation sous contraintes, les conditions d'optimalité possèdent deux parties : la faisabilité et la sensibilité. Ces deux parties nécessitent différents types de mesures expérimentales, à savoir les valeurs du critère et des contraintes, et les gradients du critère et des contraintes par rapport aux variables de décision. L'objectif de cette thèse est de développer une stratégie conceptuelle d'utilisation de ces mesures expérimentales en ligne de sorte que le procédé vérifie non seulement les conditions nécessaires, mais également les conditions suffisantes d'optimalité. Ce développement conceptuel va notamment s'appuyer sur les récents progrès en optimisation déterministe (les méthodes stochastiques ne seront pas abordées dans ce travail) de procédés basés principalement sur l'estimation des variables d'état non mesurées à l'aide d'un observateur à horizon glissant. Une méthodologie d'optimisation dynamique en temps réel (D-RTO) a été développée et appliquée à un réacteur batch dans lequel une réaction de polymérisation par greffage a lieu. L'objectif est de déterminer le profil temporel de température du réacteur qui minimise le temps opératoire tout en respectant des contraintes terminales sur le taux de conversion et l'efficacité de greffage
... The RNN model is utilized to model the batch crystallizer for FF crystallization using simulation data generated from the PBM presented in the previous section. Specifically, the RNN is constructed with input, hidden, and output layers (Figure 3), where the states in the hidden layers x ∈ R d x are represented as follows (11) where u t ∈ R d u are the RNN inputs at time t, and the weight matrices W ∈ R d x ×d u and U ∈ R d x ×d x are associated with the input and hidden state vectors, respectively. The element-wise nonlinear activation function is denoted by σ h (e.g., ReLU). ...
... The numerical sequential approaches for dynamic optimization are strongly based on the assumption that there is an available numerical procedure that is able to obtain solution trajectories satisfying the model equations. This assumption seems to be difficult to accept, because the observed increasing precision and complexity of technological processes are reflected in mathematical models, which are often highly nonlinear, unstable, numerically ill-conditioned, and possibly multi-stage [22][23][24]. Currently, simulation studies concerning the new solutions in the field of heat and mass transfer engineering indicate that an exact numerical solution of the differential-algebraic model of the system may not be possible [25]. ...
An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.
... A highly nonlinear dynamical behavior is one of the crucial features of mathematical models of real-life systems [18,19]. Especially, some characteristic physical phenomena of technological processes can be modeled by the systems of nonlinear differential-algebraic equations [2,7,28]. Recently, a dynamic development of such models of mass and heat transfer can be observed. ...
A multiple shooting based approach to an optimal control problem for highly nonlinear differential-algebraic systems (DAEs, differential-algebraic equations) is considered. The necessary optimality conditions being a consequence of the theory of variational inequalities are derived in the form of structured nonsmooth equations. A new indirect high accuracy optimization algorithm exploiting the chain structure of the mentioned equations is described. It uses the Chen-Harker-Kanzow-Smale smoothing function for the projection operator. A global superlinear (quadratic) convergence of the new algorithm is proven with using the theory of the smoothing Newton method specialized to the multiple shooting approach. The proposed algorithm is verified on the dynamic optimization of a highly nonlinear heat and mass exchange process. In general, the presented considerations have been motivated by the results of numerical simulations presented in the work Pandelidis et al., “Performance study of counter-flow indirect evaporative air coolers” Pandelidis et al. (2015).
... The use of optimal control in this process allows for reducing energy consumption and for making production more cost effective. However, application to the crystallization process of popular approaches of industrial optimal control, such as Model Predictive Control (MPC) and Nonlinear Model Predictive Control (NMPC) can be hindered by stability problems [7], [8]. Approaches based on other nonlinear control methods try also to prove global stability and to minimize the process's energy costs [9]. ...
The problem of nonlinear optimal (H-infinity) control for the industrial crystallization process is treated in this article. The dynamic model of the crystallization process undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the system a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the optimal control problem under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis.
... JModelica.org has recently been used for several optimization studies [35][36][37][38][39][40] and is also a key part of several compound tools [41][42][43][44]. JModelica.org ...
The building sector is responsible for a large part of the world's total energy use. More than half of building energy use is needed for space heating, domestic hot water heating, and space cooling. Thermal energy supply systems are used to cover these thermal energy demands and are an integral part of new buildings and neighborhoods. These systems are becoming increasingly more complex due to the inclusion of renewable energy sources and thermal storages. Advanced simulations are required to analyze the design and the operation of these complex systems in detail and are thus an important part of the transition to new and improved building energy systems.
In this work, component and system models for thermal energy supply systems were developed in the modeling language Modelica. Numerical efficiency was an important part of the development process because the aim was to analyze long periods of time. In addition, the different requirements for simulation and optimization had to be considered during model development. The models were used for dynamic simulations with Dymola as well as dynamic optimizations with JModelica.org.
The design and the operation of two case study systems were analyzed in this work: 1) an existing integrated heating and cooling system at Vulkan, Oslo and 2) a planned local district heating grid at Brøset, Trondheim. The main components of the integrated heating and cooling system at Vulkan were heat pumps, plate heat exchangers, flat plate solar collectors, water storage tanks, ice thermal energy storage, and borehole thermal energy storage. The system supplied a total floor area of 38,500m². The main components of the local district heating grid at Brøset were a heat central, distribution pipes, and customer substations. The system was assumed to supply a total floor area of 178,000m² and different system design concepts were analyzed.
... Finally, it is easy to deduce the inequality (26). Therefore, from (19), when τ = 0, one can get ...
For a class of discrete-time Markovian jump linear systems subject to operation mode disordering, a robust model predictive control method can be proposed to solve this issue. A bijective mapping scheme between the original random process and a new random process is studied to cope with the problem of operation mode disordering. At each sampling time, the original "min-max" optimization problem is transformed into a convex optimization problem with linear matrix inequalities so that the complexity of solving the optimization problem can be greatly reduced. The sufficient stability condition of the Markovian jump linear systems can be achieved by using the Lyapunov stability theory. Moreover, a state feedback control law is obtained that minimizes an infinite prediction horizon performance cost. Furthermore, the cases of uncertain and unknown transition probabilities are also considered in this study. The simulation results show that the proposed method can guarantee optimal control performance and stability of Markovian jump linear systems.
... 39 Furthermore, computationally efficient nonlinear model predictive control (NMPC) schemes for the mean length and mean aspect ratio of KDP crystals, including a discussion on the state estimation problem and the effect of uncertainties in the model, were presented. 40,41 Also, it was demonstrated experimentally that the average dimensions of a population of KDP crystals can be controlled using temperature cycling based on a pragmatic model predictive approach. 42 It is also worth noting that, in the field of protein crystallization, a MPC methodology that is concerned with crystal shape but avoids feedback of the latter by relying on growth rate models derived from extensive kinetic Monte Carlo simulations was suggested. ...
Two feedback control approaches for influencing the evolution of the average particle dimensions of populations of needle-like crystals in growth-dominated batch cooling crystallization processes are proposed. The first strategy is a path following control (PFC) approach which does not need access to kinetic models for crystal growth. The second approach consists of a considerably more complex nonlinear model predictive controller (NMPC) that requires the availability of multidimensional crystal growth rate models. The main focus lies in analyzing the effectiveness of these two controllers with respect to successfully operating the considered process, bearing in mind the differing requirements regarding the availability of kinetic models. To this end, both control strategies were coupled with a process simulation framework that features a detailed measurement model emulating the behavior of an existing monitoring device for the evolution of the particle size and shape distribution. It is demonstrated how both controllers can identify the attainable region for the average particle dimensions of a given seed population, and also reach an arbitrary target size and shape within the interior of this region. A performance benefit from operating the more complex NMPC was not observed, which renders the PFC approach suitable and sufficient for the considered application.
... Advanced fast solution algorithms are essential in terms of the application of NMPC or moving horizon estimation (MHE) in real time. Fast real time update usually increases the performance of the closed-loop optimizing control either by tackling the effect of feedback delay or by enabling faster sampling to increase optimization frequency (Zavala et al., 2008a(Zavala et al., , 2008bHuang et al., 2009;Zavala and Biegler, 2009;Wolf et al., 2011;Wolf and Marquardt, 2016;Cao et al., 2017). Note that it might sometimes be possible to find a compromise between computational time and performance between linear MPC and NMPC for steadystate processes (Gros et al., 2016). ...
The trend towards high-quality, low-volume and high-added value production has put more emphasis on semi-batch processing due to its increased flexibility of operations. Dynamic optimization plays an important role toward improving the operation of batch and semi-batch. In addition, nonlinear model predictive control (NMPC) is also an important tool for the real-time optimization of batch and semi-batch processes under uncertainty. However, the transient behaviour as well as the flexibility decrease with respect to time make the optimization of such processes very challenging.
The preferred strategy to solve constrained nonlinear dynamic optimization problems is usually to use a so-called direct method. Nevertheless, based on the problem type at hand and the solution algorithm used, direct methods may lead to computational complexity. In particular, the large prediction horizons required in shrinking-horizon NMPC increase the real-time computational effort because of expensive matrix factorizations in the solution steps, especially at the beginning of the batch. The computational delay associated with advanced control methods is usually underestimated in theoretical studies. However, this delay may contribute to suboptimal or, worse, infeasible operation in real-life applications.
Alternatively, indirect methods based on Pontryagin’s Minimum Principle (PMP) could efficiently deal with the optimization of batch and semi-batch processes. In fact, the interplay between states and co-states in the context of PMP might turn out to be computationally quite efficient. The main indirect solution technique is the shooting method, which however often leads to convergence problems and instabilities caused by the integration of the co-state equations forward in time. Generally, it has been extensively argued that indirect methods are non-convergent and inefficient for constrained problems. However, this study proposes an alternative, convergent and effective indirect solution technique. Instead of integrating the states and co-states simultaneously forward in time, the proposed algorithm parameterizes the inputs, and integrates the state equations forward in time and the co-state equations backward in time, thereby leading to a gradient-based optimization approach. Constraints are handled by indirect adjoining to the Hamiltonian function, which allows meeting the active constraints explicitly at every iteration step. The performance of the solution strategy is compared to direct methods through three different case studies. The results show that the proposed PMP-based quasi-Newton strategy is effective in dealing with complicated constraints and is quite competitive computationally.
In addition, this work suggests using the proposed indirect solution technique in the context of shrinking-horizon NMPC under uncertainty. Uncertainties can be handled by the introduction of time-varying backoff terms for the path constraints. The resulting NMPC algorithm is applied to a two-phase semi-batch reactor for the hydroformylation of 1-dodecene in the presence of uncertainty, and its performance is compared to that of NMPC that uses a direct simultaneous optimization method. The results show that the proposed algorithm (i) can enforce feasible operation for different uncertainty realizations both within batch or from batch to batch, and (ii) is significantly faster than direct simultaneous NMPC, especially at the beginning of the batch. In addition, a modification of the PMP-based NMPC scheme is proposed to enforce active constraints via tracking and reduce the real-time computational load further.
This thesis also details the combination of an indirect solution scheme together with a parsimonious input parameterization. The idea is to parameterize the sensitivity-seeking inputs in a parsimonious way so as to decrease the computational load of constrained nonlinear dynamic optimization problems. The proposed method is tested on the simulated examples of a batch binary distillation column with terminal purity constraints and a two-phase semi-batch hydroformylation reactor with a complex path constraint. The performance of the proposed indirect parsimonious solution scheme is compared with those of a fully parameterized PMP-based and a direct simultaneous solution approaches. It is observed that the combination of the indirect approach with parsimonious input parameterization can result in significant reduction in computational time.
Finally, in this work, the combination of simple solution models with parsimonious input parameterization in the context of shrinking-horizon NMPC is suggested in order to minimize the computational delay in feedback. Solution models exploit the nominal optimal solution to suggest parsimonious parameterizations (especially for sensitivity-seeking arcs) that lead to fast optimization. The proposed approach is illustrated in simulation on two case studies in the presence of uncertainty, namely a binary batch distillation column and a semi-batch reactor. The results show that the suggested parsimonious shrinking-horizon NMPC (i) performs very similarly to the standard (fully parameterized) shrinking-horizon NMPC in terms of cost, (ii) is computationally much faster than the standard shrinking-horizon NMPC especially at the beginning of the batch, (iii) is robust to plant-model mismatch
... The significant calculation time requires the application of accelerated optimization techniques. In earlier studies the most common dynamic optimization algorithms have been compared, 24,41 and it was found that the direct optimization and multiple shooting overperformed the direct single shooting in computational time; however, these were more sensitive for premature stops. Thus in this work the direct single shooting method is applied. ...
The control of batch crystallizers is an intensively investigated topic as suitable crystallizer operation can reduce considerably the downstream operation costs and produce crystals of desired properties (size, shape, purity etc.). Nevertheless, the control of crystallizers is still challenging. In this work the development of a fixed batch time full population balance model based adaptive predictive control system for cooling batch crystallizers is presented. The model equations are solved by the high resolution finite volume algorithm involving fine discretization, which provides a high fidelity, accurate solution. A physically relevant crystal size distribution (CSD) to chord length distribution (CLD) transformation is also developed making possible the direct, real-time application of the focused beam reflectance measurement (FBRM) probe in the control system. The measured CLD and concentration values are processed by the growing horizon estimator (GHE), whose roles are to estimate the un-measurable system states (CSD) and to re-adjust the kinetic parameters providing an adaptive feature for the control system. A repeated sequential optimization algorithm is developed for the nonlinear model predictive control (NMPC) optimization, enabling the reduction of sampling time to the order of minutes for the one-day long batch. According to the simulation results the strategy is highly robust to parametric plant-model mismatch and significant concentration measurement noise, providing very good control of the desired CLD.
... The control of crystallization processes in many practical cases consists in controlling the product CSD. 2,3 Since the crystallization is governed by the simultaneously ongoing nucleation and growth, which are nonlinear functions of concentration, controlling their relative rates often leads to complicated control problems. ...
In this work the modeling, simulation and optimization of an integrated batch crystallizer-wet mill system is presented. It is shown that by coupling the external wet mill to the crystallizer it is possible to increase the overall system flexibility, increases the attainable crystal size distribution (CSD) and provides a significantly better distribution shaping control, than the crystallizer alone. The population balance modelling (PBM) approach with appropriate mechanisms is applied for the description of crystal population in both the crystallizer and wet mill. This description generates a system of partial differential-integral equations, which are solved with a high resolution finite volume method, involving calculations on parallel graphical processing unit (GPU) for improved solution time. In the batch crystallizer it is assumed that primary nucleation and crystal growth are the key mechanisms, whereas in the wet mill attrition and fragmentation of crystals occurs. The nucleation and growth rate kinetics are taken from the literature and a recently developed hydrodynamic model is employed for realistic description of wet mill operation. The simulation results revealed that the simultaneous dynamic optimization of the temperature, circulation flowrate and wet mill rotation speed improve the process flexibility and lead to considerably better CSDs that can be achieved in crystallizer only configuration. The dynamic optimization also automatically discovered an unexpected optimal integrated system operation, which combines the advantages of in-situ seed generation and optimal dynamic seeding. These two features make the system suitable, theoretically, to obtain any CSD shape that is within the attainable crystal size range without the need of time-consuming, tailored seed crystal generation.
In this paper, continuous crystallization of Atorvastatin calcium (ASC) using a continuous oscillatory baffled crystallizer (COBC) has been investigated. Like most API manufacturing, ASC is manufactured batchwise and the pure API is recovered via batch combined cooling and antisolvent crystallization (CCAC) process, which has the challenges of low productivity, wide crystal size distribution (CSD) and sometimes polymorphic form contamination. To overcome the limitations of the batch crystallization, continuous crystallization of ASC was studied in a NiTech (United Kingdom) DN15 COBC, manufactured by Alconbury Weston Ltd. (AWL, United Kingdom), with the aim to improve productivity and CSD of the desired polymorph. The COBC has the advantage of high heat transfer rates and improved mixing that significantly reduces the crystallization time. It also has the advantage of spatial temperature distribution and multiple addition ports to control supersaturation and hence the crystallization process. This work uses an array of process analytical technology (PAT) tools to assess key process parameters that affect the polymorphic outcome and CSD. Two parameters were found to have significant impact on the polymorph, they are ratio of solvent to antisolvent at the point of mixing of the two streams and presence of seeds. The splitting of antisolvent into two addition ports in the COBC was found to give the desired form. The CCAC of ASC in COBC was found to be 100-fold more productive than the batch CCAC process. The cycle time for generating 100 g of desired polymorphic form of ASC also significantly reduced from 22 hours in batch process to 12 minutes in the COBC. The crystals obtained using a CCAC process in a COBC had a narrower CSD compared to that from a batch crystallization process.
In the present study, a crystallization monitoring unit consisting of an in-situ digital microscope camera and real-time image analysis is utilized for monitoring and control of a micron-sized, liquid-liquid crystallization of calcium carbonate. The crystallization process is integrated with a membrane contactor-based carbon dioxide capture process to demonstrate a sustainable CO2-to-chemical unit operation. The measurement probe transilluminates the crystal suspension and provides a live view from the crystallizer. In a series of open-loop experiments, the effects of several operating conditions such as feed flow rate and volumetric power on crystal size and particle count are investigated. For comparison purposes, solid product crystals are assessed with an offline laser diffraction technique. In the closed-loop experiments, the controlled variable is average particle diameter, and the manipulated variable is mixing intensity. The implemented set-point tracking PI controller generates actuating signals based on real-time image analysis measurement of the crystal size. Experimental results demonstrate a practical approach for measuring micron-sized particle suspensions, which is a challenge for particles with a mean diameter smaller than 15–20 μm, provides insights into the mixing intensity-based particle size controllability in fast-reaction precipitation systems and offers a framework to implement a direct design feedback control policy.
Crystallization is an economical separation and purification unit operation commonly used in the pharmaceutical industry as the last drug substance manufacturing step to obtain crystalline form active pharmaceutical ingredient (API). The quality attributes of crystallization products such as purity, crystal size, crystal shape, and polymorphic form heavily influence downstream processes and may even affect the final product performance. Crystallization is one of last missing links in end-to-end continuous pharmaceutical manufacturing development because of its complex two-phase and stochastic nature. Recent advances and remaining challenges in the development of laboratorial- and industrial-scale continuous crystallization equipment, techniques, and control strategies are discussed in this chapter accompanied by a brief overview of crystallization theories as well as process analytical technology (PAT) applications in crystallization processes. The integration of continuous crystallization is also discussed in this chapter.
Optimal control theory is applied to solve single- and multiple-objective function dynamic optimization problems for two-dimensional batch crystallization processes. Objective functions based on the product crystal aspect ratio, number of nuclei, and nucleated mass are studied. It is shown that applying a constant growth rate trajectory can minimize the product aspect ratio while early and late growth trajectories can minimize nucleated number and mass, respectively. It is also found that setting the target value of aspect ratio to be near the midpoint between the maximum and minimum feasible values permits a solution with the best values for both objectives based on nucleation simultaneously. Pareto-optimal fronts for nucleated number and mass are also plotted to understand the trade-off between objectives. The result suggests that the trade-off is stronger when value of aspect ratio constraint is near the midpoint of the feasible interval.
The gas turbine system is a major contributor of the thermal power generation process and an important complementary to the future high penetration of intermittent renewable generations. The rapidity, efficiency, and economy operation of the gas turbine system calls for advanced control algorithms and associated parameter optimization methods, which are intractable due to the strong coupling impact of complex turbine systems. In this paper, a multi-objective economic model predictive control (MOEMPC) method based on quantum simultaneous whale optimization algorithm is proposed for gas turbine system control. Firstly, the objective function of the MOEMPC strategy is formulated simultaneously considering economic indexes, terminal cost function, and stability constraints. The economic indexes reflect the change of energy consumption and throttle loss in real-time, while the terminal cost function and stability constraint guarantee the tracking accuracy under variable operation point and external disturbance. Secondly, a novel quantum simultaneous whale optimization algorithm is adopted to handle the optimization problem of MOEMPC. The introductions of the quantum coding and simultaneous search in original whale optimization algorithm have improved the convergence rate and accuracy immediately. Finally, the proposed method is applied into the controller design of gas turbine system in combined cycle unit. The simulation results have indicated the superiority of the presented strategy in terms of desired economic performance, high precision, rapidity, and robustness.
Energy systems for buildings and neighborhoods are expected to become more complex and flexible. Advanced control strategies are required to exploit the full potential of this flexibility and are especially important for systems with storages. In this study, the control of an integrated heating and cooling system for a building complex in Oslo, Norway, was analyzed. Focus was on the control setpoints for the main heat pumps, which had a total heating capacity of about 1 MW and were connected to thermal storage tanks. Previously developed simulation models of the system and its main components were made suitable for dynamic optimization with long time horizons. JModelica.org was used to find optimal control trajectories for the system with two different objectives. The first objective was to reduce the electricity use of the system and the second objective was to reduce the electricity costs of the system. The results showed that the electricity use of the system could be reduced by about 5 % for the analyzed year. The electricity costs could be reduced by about 5 to 11 % for the three analyzed winter months, depending on the variability of the electricity price and the size of the storage tanks.
Crystallization, often as the final isolation and purification step in drug substance manufacturing, has substantial impact on downstream efficiency and final drug product quality. It is a critical but challenging step in developing end-to-end continuous manufacturing, which has been identified as an emerging technology in the pharmaceutical manufacturing sector by the U.S. Food and Drug Administration (FDA). A traditional stirred tank crystallizer (STC) operated as a mixed-suspension-mixed-product-removal (MSMPR) system is a popular choice for continuous crystallization for its utilization of existing knowledge and equipment. However, there are disadvantages associated with agitational systems like the STC, such as poor local mixing and high shear rate. A commercial dynamic baffle crystallizer (DBC) was studied here as an alternative unit operation. The DBC in this study consists of a jacketed glass vessel with dynamic ‘donut’ shaped baffles to provide oscillatory mixing to improve heat and mass transfer while exerting less shear. Our goal is to compare continuous operation performances in the DBC with the STC as MSMPR systems. First, residence time distribution (RTD) studies of both systems were performed considering single phase (liquid only) and two-phase (solid-liquid) operation. The RTD gives good insights into the mixing dynamics without computationally heavy fluid dynamic simulations. Continuous cooling crystallization of paracetamol was performed in the DBC and the STC. The DBC showed good potential to be used as an MSMPR system producing more uniform RTDs as well as higher quality crystallization products compared to a traditional STC.
Monocular depth estimation is a challenging task in complex compositions depicting multiple objects of diverse scales. Albeit the recent great progress thanks to the deep convolutional neural networks (CNNs), the state-of-the-art monocular depth estimation methods still fall short to handle such real-world challenging scenarios. In this paper, we propose a deep end-to-end learning framework to tackle these challenges, which learns the direct mapping from a color image to the corresponding depth map. First, we represent monocular depth estimation as a multi-category dense labeling task by contrast to the regression based formulation. In this way, we could build upon the recent progress in dense labeling such as semantic segmentation. Second, we fuse different side-outputs from our front-end dilated convolutional neural network in a hierarchical way to exploit the multi-scale depth cues for depth estimation, which is critical to achieve scale-aware depth estimation. Third, we propose to utilize soft-weighted-sum inference instead of the hard-max inference, transforming the discretized depth score to continuous depth value. Thus, we reduce the influence of quantization error and improve the robustness of our method. Extensive experiments on the NYU Depth V2 and KITTI datasets show the superiority of our method compared with current state-of-the-art methods. Furthermore, experiments on the NYU V2 dataset reveal that our model is able to learn the probability distribution of depth.
Representing the uncertainties with a set of scenarios, the optimization problem resulting from a robust nonlinear model predictive control (NMPC) strategy at each sampling instance can be viewed as a large-scale stochastic program. This paper solves these optimization problems using the parallel Schur complement method developed to solve stochastic programs on distributed and shared memory machines. The control strategy is illustrated with a case study of a multidimensional unseeded batch crystallization process. For this application, a robust NMPC based on min–max optimization guarantees satisfaction of all state and input constraints for a set of uncertainty realizations, and also provides better robust performance compared with open-loop optimal control, nominal NMPC, and robust NMPC minimizing the expected performance at each sampling instance. The performance of robust NMPC can be improved by generating optimization scenarios using Bayesian inference. With the efficient parallel solver, the solution time of one optimization problem is reduced from 6.7 min to 0.5 min, allowing for real-time application.
We present a clustering-based preconditioning strategy for KKT systems arising in stochastic programming within an interior-point framework. The key idea is to perform adaptive clustering of scenarios (inside-the-solver) based on their influence on the problem at hand. This approach thus contrasts with existing (outside-the-solver) approaches that cluster scenarios based on problem data alone. We derive spectral and error properties for the preconditioner and demonstrate that scenario compression rates of up to 94 % can be obtained, leading to dramatic computational savings. In addition, we demonstrate that the proposed preconditioner can avoid scalability issues of Schur decomposition in problems with large first-stage dimensionality.
This paper develops a stochastic hybrid model-based control system that can determine online the optimal control actions, detect faults quickly in the control process, and reconfigure the controller accordingly using interacting multiple-model (IMM) estimator and generalized predictive control (GPC) algorithm. A fault detection and control system consists of two main parts: the first is the fault detector and the second is the controller reconfiguration. This work deals with three main challenging issues: design of fault model set, estimation of stochastic hybrid multiple models, and stochastic model predictive control of hybrid multiple models. For the first issue, we propose a simple scheme for designing faults for discrete and continuous random variables. For the second issue, we consider and select a fast and reliable fault detection system applied to the stochastic hybrid system. Finally, we develop a stochastic GPC algorithm for hybrid multiple-models controller reconfiguration with soft switching signals based on weighted probabilities. Simulations for the proposed system are illustrated and analyzed.
The demand for fast solution of nonlinear optimization problems, coupled with the emergence of new concurrent computing architectures, drives the need for parallel algorithms to solve challenging nonlinear programming (NLP) problems. In this paper, we propose an augmented Lagrangian interior-point approach for general NLP problems that solves in parallel on a Graphics processing unit (GPU). The algorithm is iterative at three levels. The first level replaces the original problem by a sequence of bound-constrained optimization problems using an augmented Lagrangian method. Each of these bound-constrained problems is solved using a nonlinear interior-point method. Inside the interior-point method, the barrier sub-problems are solved using a variation of Newton's method, where the linear system is solved using a preconditioned conjugate gradient (PCG) method, which is implemented efficiently on a GPU in parallel. This algorithm shows an order of magnitude speedup on several test problems from the COPS test set.
In this work, we address optimization of large-scale, nonlinear, block-structured problems with a significant number of coupling variables. Solving these problems using interior-point methods requires the solution of a linear system that has a block-angular structure at each iteration. Parallel solution is possible using a Schur-complement decomposition. In an explicit Schur-complement decomposition, the computational cost of forming and factorizing the Schur-complement is prohibitive for problems with many coupling variables. In this paper, we show that this bottleneck can be overcome by solving the Schur-complement equations implicitly, using a quasi-Newton preconditioned conjugate gradient method. This new algorithm avoids explicit formation and factorization of the Schur-complement. The computational efficiency of this algorithm is compared with the serial full-space approach, and the serial and parallel explicit Schur-complement approach. These results show that the PCG Schur-complement approach dramatically reduces the computational cost for problems with many coupling variables.
The crystal product quality, functionality and properties such as flow characteristics are greatly influenced by the crystal size and shape distributions. Hence, prediction and control of crystal size and shape distributions are important from the viewpoint of smooth operation and product quality control. The contribution of this paper is twofold: firstly, it describes the modelling and simulation aspects for prediction of crystal shape distribution in the presence of crystal growth modifiers (CGMs). Secondly, a control setup is proposed that manipulates the CGM concentration in the crystallizer in order to achieve the desired crystal shape. The effect of CGM on crystal shape is modelled using a morphological population balance model (PBM) in combination with Kubota-Mullin's model for the pinning mechanism of CGM molecules on the crystal face to take into account their possible hindering effect on crystal growth (Kubota and Mullin, 1995; Kubota, 2001). Subsequently, a simple feedback control configuration is implemented in the proposed control setup which manipulates the CGM concentration profile in the crystallizer. The control setup uses a hybrid continuous-batch operation of the crystallizer, in which the solid phase is operated in a batch mode, whereas the liquid phase is operated in a continuous mode maintaining a constant volume in the reactor. This hybrid operation allows the dynamic control of the CGM concentration during the batch. Using potassium dihydrogen phosphate (KDP) as a model system, the simulation results demonstrate that, in principle, the crystal shape distribution can be controlled by manipulating the CGM concentration profile.
The optimization of unseeded batch cooling crystallization systems is studied in a multi-objective framework. The length mean size and target aspect ratio (AR) of final crystals were considered as objectives, which give rise to a set of optimal solutions known as Pareto-optimal solutions. Two model compounds were considered: paracetamol and potassium dyhidrogen phosphate (KDP), nucleation and growth dominated systems, respectively. The optimization of KDP showed a dependence between the optimal profile and the target objectives weights. The optimal profile for paracetamol did not vary from a cubic cooling profile due to the low sensitivity of the AR to temperature changes. Furthermore, the longer batch time has a greater impact in the optimal AR that could be achieved due to the variation in supersaturation dependence. Moreover, it was shown that there is a relation between the classification of crystallization kinetics based on growth and nucleation mechanisms and the set of optimal cooling profiles for shape and size optimization.
The purpose of the current work is to develop a systematic classification scheme for crystallization systems considering simultaneous size and shape variations, and to study the effect of temperature profiles on the achievable final shape of crystals for various crystallization systems. A classification method is proposed based on the simultaneous consideration of the effect of temperature profiles on nucleation and growth rates of two different characteristic crystal dimensions. Hence the approach provides direct indication of the extent in which crystal shape may be controlled for a particular system class by manipulating the supersaturation. A multidimensional population balance model (PBM) was implemented for unseeded crystallization processes of four different compounds. The effect between the nucleation and growth mechanisms on the final aspect ratio (AR) was investigated and it was shown that for nucleation dominated systems the AR is independent of the supersaturation profile. The simulation results confirmed experimentally also show that most crystallization systems tend to achieve an equilibrium shape hence the variation in the aspect ratio that can be achieved by manipulating the supersaturation is limited, in particular when nucleation is also taken into account as a competing phenomenon.
Given that the fundamental process of crystal growth and its associated kinetic control is surface controlled, the use of a single scalar parameter, particle size, usually defined as a volume equivalent diameter, i.e., based on a spherical assumption of particle shape can be misleading for a number of practical crystallization systems, notably pharmaceutical products. Hence, measurement of the growth rate for each individual crystal surface in real-time and within processing reactors could open the way for the development of more effective process and concomitant product quality control. This paper presents the measurement of the growth rates of needle-shaped crystals in two dimensions using on-line imaging and image analysis techniques through a feasibility study of the batch crystallization of β form L-glutamic acid. The length and width of each needle-shaped crystal were measured every 60 s, ranging from 100 to nearly 180 μm in length and from 30 to 45 μm in width, and the values were used to estimate growth rates on both directions. The growth rate in length was found to be four to six times greater than for the width. The (101) plane was found to be the fastest growing surface of the morphology studied and an attempt has been made to estimate its growth-kinetics parameters from measurements of length, whilst it was harder to estimate kinetics from measurements of width for other crystal facets.
Crystallization is one of the most important unit operations used for the separation and purification of crystalline solid products. Appropriate design and control of the crystallization process is paramount to produce crystalline products with tailor-made-properties. This paper provides an overview of selected recent developments in the modelling, monitoring and control of crystallization processes. We consider the topics discussed in this review to be enabling technologies for the development of the next generation of crystallization processes with significantly improved predictability, robustness and controllability.
Advances in sensor technology and increased competition in the pharmaceutical industry have generated significant interest in the identification of models for the solution formation of crystals with multiple characteristic dimensions. A procedure is proposed that uses a small number of batch experiments to identify the kinetic parameters for multidimensional crystallization processes. The parameters are estimated simultaneously from the on-line measurement of infrared spectra and from cross-moments of the crystal size distribution. The identification procedure maximizes the informativeness of the data produced by each experiment, produces an estimate of the accuracy of the kinetic parameters, and allows the consideration of competing hypotheses for characterizing the crystallization kinetics. The parameter identification strategy is applied to the batch crystallization of potassium dihydrogen phosphate, which forms two-dimensional crystal from solution. To the best of the author's knowledge, this is the first time that the kinetic parameters for a multidimensional crystallization process are identified from a small number of batch experiments.
NMPC explicitly addresses constraints and nonlinearities during the feedback control of batch processes. This NMPC algorithm also explicitly takes parameter uncertainty into account in the state estimation and state feedback controller designs. An extended Kalman filter estimates the process noise co®ariance matrix from the parameter uncer- tainty description and employs a sequential integration and correction strategy to reduce biases in the state estimates due to parameter uncertainty. The shrinking horizon NMPC algorithm minimizes a weighted sum of the nominal performance objecti®e, an estimate of the ®ariance of the performance objecti®e, and an integral of the de®iation of the control trajectory from the nominal optimal control trajectory. The robust performance is quantified by estimates of the distribution of the performance index along the batch run obtained by a series expansion about the control trajectory. The control and analysis approaches are applied to a simulated batch crystallization process with a realistic un- certainty description. The proposed robust NMPC algorithm impro®es the robust perfor- mance by a factor of six compared to open loop optimal control, and a factor of two compared to nominal NMPC. Monte Carlo simulations support the results obtained by the distributional robustness analysis technique.
Abstract The goal of state estimation,is to reconstruct,the state of a system,from,process,mea- surements,and,a model.,State estimators,for most,physical,processes,often,must,ad- dress many different challenges, including nonlinear dynamics, states subject to hard constraints (e.g. nonnegative concentrations), and local optima. In this article, we com- pare,the performance,of two,such,estimators:,the,extended,Kalman,filter (EKF) and moving,horizon,estimation,(MHE). We illustrate conditions,that lead to estimation,fail- ure in the EKF when,there is no plant-model,mismatch,and,demonstrate,such,failure via several simple examples. We then examine the role that constraints, the arrival cost, and the type,of optimization,(global versus,local) play,in determining,how,MHE performs on these examples. In each example, the two estimators are given exactly the same in- formation, namely tuning parameters, model, and measurements; yet MHE consistently provides,improved,state estimation,and,greater,robustness,to both,poor,guesses,of the initial state and,tuning,parameters,in comparison,to the EKF.
This article presents a model-based control approach for optimal operation of a seeded fed-batch evaporative crystallizer. Various direct optimization strategies, namely, single shooting, multiple shooting, and simultaneous strategies, are used to examine real-time implementation of the control approach on a semi-industrial crystallizer. The dynamic optimizer utilizes a nonlinear moment model for on-line computation of the optimal operating policy. An extended Luenberger-type observer is designed to enable closed-loop implementation of the dynamic optimizer. In addition, the observer estimates the unmeasured process variable, namely, the solute concentration, which is essential for the intended control application. The model-based control approach aims to maximize the batch productivity, as satisfying the product quality requirements. Optimal control of crystal growth rate is the key to fulfill this objective. This is due to the close relation of the crystal growth rate to product attributes and batch productivity. The experimental results suggest that real-time application of the control approach leads to a substantial increase, i.e., up to 30%, in the batch productivity. The reproducibility of batch runs with respect to the product crystal size distribution is achieved by thorough seeding. The simulation and experimental results indicate that the direct optimization strategies perform similarly in terms of optimal process operation. However, the single shooting strategy is computationally more expensive. © 2010 American Institute of Chemical Engineers AIChE J, 57: 1557–1569, 2011
A population balance model for predicting the dynamic evolution of crystal shape distribution is further developed to simulate crystallization processes in which multiple crystal morphological forms co-exist and transitions between them can take place. The new model is applied to derive the optimal temperature and supersaturation profiles leading to the desired crystal shape distribution in cooling crystallization. Since tracking an optimum temperature or supersaturation trajectory can be easily implemented by manipulating the coolant flowrate in the reactor jacket, the proposed methodology provides a feasible closed-loop mechanism for crystal shape tailoring and control. The methodology is demonstrated by applying it to a case study of seeded cooling crystallization of potash alum. © 2009 American Institute of Chemical Engineers AIChE J, 2009
We present a simple technique that can identify suitable data from a noisy signal produced on-line by commercially available image-analysis software. A controller successfully uses this signal to regulate the flow rate of a habit modifier stream to maintain a desired crystal habit. We demonstrate these methods on a simple chemical system: sodium chlorate (NaClO 3 ) crystallization using sodium dithionate (Na 2 S 2 O 6 ) as a habit modifier.
Crystallization processes in the pharmaceutical industry are usually designed to obtain crystals with controlled size, shape, purity, and polymorphic form. Knowledge of the process conditions required to fabricate crystals with controlled characteristics is critical during process development. In this work, continuous crystallization of ketoconazole, flufenamic acid, and l-glutamic acid in a nonconventional plug flow crystallizer was investigated. Kenics type static mixers were used to promote homogeneous mixing of active pharmaceutical ingredient solution and antisolvent. A strategy of multiple points of addition of antisolvent along the crystallizer was evaluated to control the size of the crystals. Interestingly, it was found that crystal size can be increased or decreased with an increased number of antisolvent addition points, depending on the kinetics of the system. It was also found that smaller crystals with a narrower size distribution can be obtained with the static mixers. A model to describe the continuous crystallization process was developed through the simultaneous solution of a population balance equation, kinetics expressions for crystal growth and nucleation, and a mass balance. The comparison of experimental and calculated values for crystal size distribution revealed that a growth rate dispersion model could describe accurately the continuous crystallization process. Collision of crystals with each other and with mixing elements inside the crystallizer may be the source of random fluctuation of the growth rate in the nonconventional plug flow crystallizer with static mixers.
This paper presents an output feedback nonlinear model-based control approach for optimal operation of industrial batch crystallizers. A full population balance model is utilized as the cornerstone of the control approach. The modeling framework allows us to describe the dynamics of a wide range of industrial batch crystallizers. In addition, it facilitates the use of performance objectives expressed in terms of crystal size distribution. The core component of the control approach is an optimal control problem, which is solved by the direct multiple shooting strategy. To ensure the effectiveness of the optimal operating policies in the presence of model imperfections and process uncertainties, the model predictions are adapted on the basis of online measurements using a moving horizon state estimator. The nonlinear model-based control approach is applied to a semi-industrial crystallizer. The simulation results suggest that the feasibility of real-time control of the crystallizer is largely dependent on the discretization coarseness of the population balance model. The control performance can be greatly deteriorated due to inadequate discretization of the population balance equation. This results from structural model imperfection, which is effectively compensated for by using the online measurements to confer an integrating action to the dynamic optimizer. The real-time feasibility of the output feedback control approach is experimentally corroborated for fed-batch evaporative crystallization of ammonium sulphate. It is observed that the use of the control approach leads to a substantial increase, i.e., up to 15%, in the batch crystal content as the product quality is sustained.
The optimal batch control of a multidimensional crystallization process is investigated. A high resolution algorithm is used to simulate the multidimensional crystal size distribution under the operations defined by two optimal control trajectories. It is shown that a subtle change in the optimal control objective can have a very large effect on the crystal size and shape distribution of the product crystals. The effect of spatial variation is investigated using a compartmental model. The effect of differing numbers of compartments on the size and shape distribution of the product crystals is investigated. It is shown that the crystal size distribution can be very different along the height of the crystallizer and that a solution concentration gradient exists due to imperfect mixing. The nucleation rate can be significantly larger at the bottom of the crystallizer and the growth rate can be much larger at the top. The high resolution method provides high simulation accuracy and fast speed, with the ability to solve large numbers of highly nonlinear coupled multidimensional partial differential equations over a wide range of length scales. A parallel programming implementation results in simulation times that are short enough for using the simulation program to compute optimal control trajectories.
In order to obtain constant solid properties with particles exhibiting a low order of symmetry, it is necessary to monitor and to control several distributed parameters characterising the crystal shape and size. A bi-dimensional population balance model was developed to simulate the time variations of two characteristic sizes of crystals. The nonlinear population balance equations were solved numerically over the bi-dimensional size domain using the so-called method of classes. An effort was made to improve usual simulation studies through the introduction of physical knowledge in the kinetic laws involved during nucleation and growth phenomena of complex organic products. The performances of the simulation algorithm were successfully assessed through the reproduction of two well-known theoretical and experimental features of ideal continuous crystallization processes: the computation of size-independent growth rates from the plot of the steady-state crystal size distribution and the possibility for MSMPR crystallizers to exhibit low-frequency oscillatory behaviours in the case of insufficient secondary nucleation.
A multivariable multi-rate nonlinear model predictive control (NMPC) strategy is applied to styrene polymerization. The NMPC algorithm incorporates a multi-rate Extended Kalman Filter (EKF) to handle state variable and parameter estimation. A fundamental model is developed for the styrene polymerization CSTR, and control of polymer properties such as number average molecular weight (NAMW) and polydispersity is considered. These properties characterize the final polymer distribution and are strong indicators of the polymer qualities of interest. Production rate control is also demonstrated. Temperature measurements are available frequently while laboratory measurements of concentration and molecular weight distribution are available infrequently with substantial time delays between sampling and analysis. Observability analysis of the augmented system provides guidelines for the design of the augmented disturbance model for use in estimation using the multi-rate EKF. The observability analysis links measurement sets and corresponding observable disturbance models, and shows that measurements of moments of the polymer distribution are essential for good estimation and control. The CSTR is operated at an open-loop unstable steady state. Control simulations are performed under conditions of plant-model structural mismatch and in the presence of parameter uncertainty and disturbances, and the proposed multi-rate NMPC algorithm is shown to provide superior performance compared to linear multi-rate and nonlinear single-rate MPC algorithms. The major contributions of this work are the development of the multi-rate estimator and the measurement design study based on the observability analysis.
Following on the popularity of dynamic simulation for process systems, dynamic optimization has been identified as an important task for key process applications. In this study, we present an improved algorithm for simultaneous strategies for dynamic optimization. This approach addresses two important issues for dynamic optimization. First, an improved nonlinear programming strategy is developed based on interior point methods. This approach incorporates a novel filter-based line search method as well as preconditioned conjugate gradient method for computing search directions for control variables. This leads to a significant gain in algorithmic performance. On a dynamic optimization case study, we show that nonlinear programs (NLPs) with over 800,000 variables can be solved in less than 67 CPU minutes. Second, we address the problem of moving finite elements through an extension of the interior point strategy. With this strategy we develop a reliable and efficient algorithm to adjust elements to track optimal control profile breakpoints and to ensure accurate state and control profiles. This is demonstrated on a dynamic optimization for two distillation columns. Finally, these algorithmic improvements allow us to consider a broader set of problem formulations that require dynamic optimization methods. These topics and future trends are outlined in the last section.
Simultaneous approaches for dynamic optimization problems are surveyed and a number of emerging topics are explored. Also known as direct transcription, this approach has a number of advantages over competing dynamic optimization methods. Moreover, a number of industrial applications have recently been reported on challenging real-world applications. This study provides background information, summarizes the underlying concepts and properties of this approach, discusses recent advances in the treatment of discrete decisions and, finally, illustrates the approach with two process case studies.
A key bottleneck in the production of pharmaceuticals and many other products is the formation of crystals from solution. The control of the crystal size distribution can be critically important for efficient downstream operations such as filtration and drying, and product effectiveness (e.g., bioavailability, tablet stability). This paper provides an overview of recent developments in the control of crystallization processes, including activities in sensor technologies, model identification, experimental design, process simulation, robustness analysis, and optimal control.
Crystallization is the main separation and purification process for the manufacturing of drug substances. Not only does crystallization affect the efficiency of downstream operations such as filtering, drying, and formulating, the efficacy of the drug can be dependent on the final crystal form. Advances in simulation and control algorithms and process sensor technologies have enabled the development of systematic first-principles and direct design approaches for the batch control of crystallization processes. These approaches address different challenges associated with pharmaceutical crystallization control. This paper provides an overview of recent technological advances in the in situ control of pharmaceutical crystallization processes. Implementation of the first-principles and direct design approaches are compared, and their relative merits are explained. Areas of future opportunities for application of advanced control strategies in pharmaceutical crystallization are presented.
The Modelica language, targeted at modeling of complex physical systems, has gained increased attention during the last decade. Modelica is about to establish itself as a de facto standard in the modeling community with strong support both within academia and industry. While there are several tools, both commercial and free, supporting simulation of Modelica models few efforts have been made in the area of dynamic optimization of Modelica models. In this paper, an extension to the Modelica language, entitled Optimica, is reported. Optimica enables compact and intuitive formulations of optimization problems, static and dynamic, based on Modelica models. The paper also reports a novel Modelica-based open source project, JModelica.org, specifically targeted at dynamic optimization. JModelica.org supports the Optimica extension and offers an open platform based on established technologies, including Python, C, Java and XML. Examples are provided to demonstrate the capabilities of Optimica and JModelica.org.
The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.
The fundamental processes of crystal nucleation and growth are strongly dependent on the solute concentration. A significant limitation to the development of reliable techniques for the modeling, design, and control of crystallization processes has been the difficulty in obtaining highly accurate supersaturation measurements for dense suspensions. Attenuated total reflection Fourier transform infrared spectroscopy is coupled with chemometrics to provide highly accurate in situ solute concentration measurement in dense crystal slurries. At the 95% confidence level, the chemometric techniques provided solute concentration estimates with an accuracy of ±0.12 wt% for potassium dihydrogen phosphate.
This paper provides an overview of currently available methods for state estimation of linear, constrained and nonlinear systems. The following methods are discussed: Kalman filtering, extended Kalman filtering, unscented Kalman filtering, particle filtering, and moving horizon estimation. The current research literature on particle filtering and moving horizon estimation is reviewed, and the advantages and disadvantages of these methods are presented. Topics for new research are suggested that address combining the best features of moving horizon estimation and particle filters.
Model predictive control is a form of control in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state; the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. An important advantage of this type of control is its ability to cope with hard constraints on controls and states. It has, therefore, been widely applied in petro-chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls. In this review, we focus on model predictive control of constrained systems, both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and/or time-varying systems. We concentrate our attention on research dealing with stability and optimality; in these areas the subject has developed, in our opinion, to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control. We distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature. In some cases the finite horizon optimal control problem solved on-line is exactly equivalent to the same problem with an infinite horizon; in other cases it is equivalent to a modified infinite horizon optimal control problem. In both situations, known advantages of infinite horizon optimal control accrue.
This study formulates a class of problems in particle technology in terms of equations familiar from classical statistical mechanics, and shows how these equations can be tied in to the differential material and energy balances commonly used to describe the performance of pieces of chemical processing equipment. The main problems treated are those of particle nucleation and growth, and, weakly, agglomeration.
This paper provides an overview of commercially available model predictive control (MPC) technology, both linear and nonlinear, based primarily on data provided by MPC vendors. A brief history of industrial MPC technology is presented first, followed by results of our vendor survey of MPC control and identification technology. A general MPC control algorithm is presented, and approaches taken by each vendor for the different aspects of the calculation are described. Identification technology is reviewed to determine similarities and differences between the various approaches. MPC applications performed by each vendor are summarized by application area. The final section presents a vision of the next generation of MPC technology, with an emphasis on potential business and research opportunities.
In this work, an efficient numerical method is introduced for solving one-dimensional batch crystallization models with size-dependent growth rates. The proposed method consist of two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for the moments and the solute concentration is numerically solved to obtain their discrete values in the time domain of interest. These discrete values are also used to get growth and nucleation rates in the same time domain. To overcome the issue of closure, a Gaussian quadrature method based on orthogonal polynomials is employed for approximating integrals appearing in the ODE system. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The expression for CSD is obtained by applying the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate, and easy to implement in the computer. Several numerical test problems of batch crystallization processes are considered. For a validation, the results of the proposed technique are compared with those obtained using a high resolution finite volume scheme. Copyright © 2009 Elsevier Ltd All rights reserved. [accessed July 24, 2009]
The fundamental processes of crystal nucleation and growth are
strongly dependent on the concentration of the solute in solution. A
significant limitation to the development of reliable rigorous
techniques for the modeling, design, and control of crystallization
processes has been the difficulty in obtaining highly accurate
supersaturation measurements for dense suspensions. Attenuated total
reflectance (ATR) Fourier transform infrared (FTIR) spectroscopy is
coupled with robust chemometrics analysis to provide highly accurate
online supersaturation estimation in dense crystal slurries.
Supersaturation estimates constructed from robust chemometrics
techniques are substantially more accurate than using the conventional
approaches for FTIR data analysis
The paper provides a reasonably accessible and self-contained
tutorial exposition on model predictive control (MPC). It is aimed at
readers with control expertise, particularly practitioners, who wish to
broaden their perspective in the MPC area of control technology. We
introduce the concepts, provide a framework in which the critical issues
can be expressed and analyzed, and point out how MPC allows
practitioners to address the trade-offs that must be considered in
implementing a control technology
State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.