Control strategy for biopharmaceutical production by model predictive control

The biopharmaceutical industry is rapidly advancing, driven by the need for cutting‐edge technologies to meet the growing demand for life‐saving treatments. In this context, Model Predictive Control (MPC) has emerged as a promising solution to address the complexity of modern biopharmaceutical production processes. Its ability to optimize operations and ensure consistent product yields has made it an attractive option for manufacturers in this sector. Furthermore, MPC's alignment with the Process Analytical Technology (PAT) initiative provides an additional layer of assurance, facilitating real‐time monitoring and enabling swift adjustments to maintain process integrity. This comprehensive review delves into the various applications of MPC, ranging from robust control to stochastic model predictive control, thereby equipping biotechnologists and process engineers with a powerful toolset. By harnessing the capabilities of MPC, as elucidated in this review, manufacturers can confidently navigate the intricate bioprocessing landscape and unlock this approach's full potential in their production processes.


| INTRODUCTION
Controlling biopharmaceutical manufacturing systems is a formidable undertaking that presents numerous challenges for scientists and engineers alike. 1 The increasing complexity of biological systems and the growing demand for precision in drug development and manufacturing necessitate the development of advanced process control methods that can ensure consistent product yields and highquality outputs.2][13][14] Despite its proven track record, the application of MPC in biopharmaceutical production has been limited due to the unique challenges posed by biological systems and the requirement for advanced sensors capable of measuring critical quality attributes.
Biological systems are characterized by their high complexity and dynamic behavior, 15 rendering them difficult to control using conventional process control methods. 16The quality of the final biopharmaceutical product is influenced by a multitude of factors, including cell growth rate, impurity presence, growth media composition, downstream processing, hold steps, and raw material quality.These factors can undergo rapid and unpredictable changes, making it challenging to maintain consistent product quality and yields.In response to these challenges, the US Food and Drug Administration (FDA) initiated the Process Analytical Technology (PAT) in 2004. 17[20] Consequently, new guidelines for continuous biomanufacturing have been established, emphasizing the utilization of advanced process control methods such as MPC. 21,22ntinuous biomanufacturing represents a novel pharmaceutical production approach involving interrupted material processing and real-time monitoring of critical process parameters to ensure consistent product quality and yield. 23,24This approach offers numerous advantages over traditional batch processing methods, including increased efficiency, greater flexibility, and reduced costs. 25However, while continuous biomanufacturing has been implemented in various capacities, the use of advanced process control methods like MPC can significantly enhance its effectiveness, especially in handling the intricate dynamics of biological systems and ensuring precise control over critical process parameters. 26Furthermore, advancements in sensor technology and real-time monitoring have enabled the continuous monitoring of critical process parameters, facilitating rapid adjustments to optimize process parameters and ensure consistent product quality and yields. 23is review provides an in-depth exploration of MPC, encompassing its principles, governing equations, and applications in the field of biotechnology.It serves as a valuable resource for biotechnologists and bioprocess engineers seeking to implement MPC in their processes, offering insights and guidance to harness the full potential of advanced process control in biopharmaceutical manufacturing.

| MODEL PREDICTIVE CONTROL
Model predictive control stands for a model-based control technique, as its name indicates.Initially, MPC employs the mathematical model describing a system to anticipate its future behavior within various conditions and uncertainties. 27,28Then, it optimizes the system by designing a control signal sequence (u k ), to steer the process to an optimal condition considering the objective function and constraints, Figure 1.Moreover, MPC strategies can be applied to multi-input and multi-output systems (MIMO) concerning equality and inequality constraints on input and output variables. 29In fact, incorporating inequality constraints was a key driving force behind the early development of MPC. 30,31Input constraints are related to the physical limitations of the system, such as the maximum power of a pump, and output constraint depends on the desired outcome of the process.
Optimization problems can be solved in many ways (e.g., genetic algorithm). 4,32,33However, they can be classified into two main categories, derivative-free and derivative-based optimization. 34,35The solver choice is based on the structure and complexity of the problem, such as linearity or non-linearity of the cost function and constraints.
Derivative-free algorithms explore the system statistically, and the derivative-based algorithms use the derivative of the system to locate the local/global optimum solution.
Indisputably, the modeling process is the main principle of MPC design, as authors 36,37 have stated this fact: "The effectiveness of any feedback design is fundamentally limited by system dynamics and model accuracy".Thus, a good optimization can be expected if the process model properly reflects the system's dynamics.However, the model description must be computationally tractable for rendering online optimization. 38Let us consider a linear time-invariant (LTI) model of a system with a discrete format: where x R nx , u R nu , and y R ny are the state, control input, and measurement at timing step k.A R nxÂnx , B R nxÂnu , and C R ny Ânx are the state-space transformation matrices.n x , n y , and n u are the dimension of x, y, and u respectively.The following optimal control problem (OCP) must be solved by MPC at each sampling time k: where J Ã is the optimal value of the cost function J x k ,u k ð Þ, N p is the prediction horizon, and the subscript k þ i j k denotes the predicted value of a variable at timing step k þ i regarding the current knowledge of the system at timing step k: 39 Equations (3b)-(3e) denote the hard constraints over the OCP, whereas Equations (3c), (3d) are the equality and inequality constraints, respectively.Hard constraints are those for which satisfaction is mandatory. 40,41Equation (3e) is an inequality constraint for the control signal (u).G R nsÂnx and H R nsÂnx are the transformation matrix corresponding to the constraints of the n s state, and D R niÂnu denotes the transformation matrix of n i control input constraint.g, h, and d are constants for each of the constraints.As illustrated in Figure 2, MPC encompasses various types, principles, and applications, demonstrating its versatility and adaptability across different industries, as discussed in the following sections.

| Cost function
The kernel of MPC is the cost function, also known as the objective function (Equation (3a)).Derivation of the cost function depends on the process's objective and desired behavior of the controlled system.
Similarly, the performance of the system is also derived by the cost function, such as minimizing the buffer consumption and maximizing the product purity.In addition, the cost function serves to stabilize the system if the optimal cost can be formed as a Lyapunov function. 423][54][55][56] In this approach, the controller forces the system to reach and follow a time-varying trajectory with a gradual transition. 57In this case, the cost function at Equation (3a) can be rewritten as: The structural overview of model predictive control (MPC), including its types, principles, uncertainties, tuning parameters, and applications across various industries.
where Δu k indicates the increments of the control action compared to the last timing step (k À 1).N P and N C are the prediction and control horizon, respectively.y kþ1 and r kþ1 are the predicted output and reference to be followed over the prediction horizon respectively, where the difference between these two corresponds to the prediction error.
Q and R are the matrices for weighting the future error and control output, respectively.In addition, Q is a positive definite matrix, and R is a positive semi-definite matrix.
Goal attainment is a powerful method for solving multi-objective optimization problems [58][59][60][61] initially introduced by Gembicki et al. 62 The vast majority of MPC problems employ the goal attainment method for finding the best compromise between weighted goals.
Using this technique often leads to solutions near the global optimum points.In this method, a set of goals is expressed by that is related to a set of objective functions g , and let x be the vector of optimizing variables.Again, each goal is weighted to demonstrate its priority over the other goals, which is represented by vector w ¼ w 1 , w 2 , …, w n f g .
Accordingly, the new objective function is: where the λ is the slackness variable, which must be minimized for all the objectives.If the weights are non-zero (w ≠ 0), then the problem alters to an optimization problem with soft constraints.Accordingly, this formulation allows the objectives to be under or overachieved. 58nimax: This technique, as its name indicates, aims to minimize the maximum value of a set of decision variables, also known as Min-Max.4][65][66][67][68][69][70][71][72] Furthermore, minimax cost functions are typically used to find conservative solutions of optimization problems in the presence of uncertainties. 41,73

| Pareto front
The concept of the Pareto front, named after economist Vilfredo Pareto, provides a method for identifying practical solutions to multiobjective problems. 746][77][78][79] By utilizing the Pareto front, decision-makers can navigate the trade-offs between different objectives and make informed choices that optimize overall performance.This approach allows for a comprehensive exploration of the solution space, enabling the identification of non-dominated solutions that represent the best compromise between conflicting objectives.
In the model predictive control context, Pareto front optimization emerges as a powerful technique for simultaneously optimizing multiple objectives.For instance, in MPC applications, one might aim to maximize production while minimizing the consumption of raw materials. 80However, these objectives often present inherent conflicts, as increasing one may lead to a decrease in the other. 81By leveraging the Pareto front approach, control designers gain a systematic and rigorous means of exploring the trade-offs between these conflicting objectives. 82This enables them to make informed decisions and provides a framework for identifying and selecting non-dominated solutions representing the most effective compromises, thereby driving enhanced performance and efficiency in various domains.
The set of non-dominated solutions, known as the Pareto front, takes the form of a plane curve, as depicted in Figure 3.It is important to note that achieving the ideal point, which represents the best possible values for all objectives with simultaneous minimization or maximization, is typically unattainable due to conflicting objectives.
Therefore, the optimal solution is determined based on its proximity to the ideal point.
For more in-depth information on the Pareto front and its applications, readers may refer to the study by Giagkiozis et al. 83 This reference provides additional insights and a comprehensive understanding of the topic.

| Tuning parameters
Besides the choice of MPC strategy, several essential parameters directly impact the performance of the MPC algorithm, including prediction and control horizon (N p and N C ), sampling interval, and weighting matrices The schematic of the Pareto front for finding the optimal solution to a problem with two objectives.
(Q and R in Equation 4).These constant factors must be adequately tuned to guarantee the optimality and robustness of the controlled system.However, the first step in the "tuning process" is developing a satisfactory process model.Subsequently, if the model is precise enough, the remaining tuning steps will be simple.In the same sense, if the controller performs poorly, it is reasonable to assume that the model is inaccurate unless proven otherwise. 31In particular, a high performance MPC requires a high precision mathematical model.5][86] Mammalian cells, for instance, are complex entities with intricate cellular pathways and behaviors, demanding a nuanced approach for accurate modeling.On the other hand, microbial models like E.coli and yeast are generally more straightforward, though not without their own intricacies.This distinction further emphasizes the pivotal role of model accuracy in MPC, as the complexity of the system being controlled can significantly influence the control strategy's effectiveness.
As explained, the objective function may contain more than one goal; in such cases, the weighting factors prioritize between the multiple goals, usually conflicting. 87The weighting factors in multi-objective situations are often discovered by examining the trade-off surfaces or curves between the objectives.However, since selecting acceptable values for the weighting factors is noticeably tedious and challenging, MPC designers frequently tend to resort to empirical strategies that depend on trial-and-error techniques. 28,87,880][91][92][93] For instance, Machado et al. 89  In this review, we have provided guidelines to avoid unnecessary and laborious trial-error processes in choosing the proper value since these parameters are unique for each task.
Prediction horizon (N p ): The prediction horizon defines the upper limit of the future prediction, by which the controller tries to induce the desired output to the system. 31It can be finite or infinite, in which case it should be correctly adjusted using tuning criteria to guarantee closed-loop stability and robustness.Moreover, it must be long enough to capture the control system's transient dynamics. 3,94While increasing the prediction horizon can lead to a less aggressive control system and generally improve performance, it also increases the computational effort exponentially.Karamanakos et al. 95 and Geyer et al. 96 have outlined the benefit of a long prediction horizon.Wojsznis et al. 97 proposed increasing the prediction horizon until further increment has no discernible impact on control performance.Garcia et al. 30 presented a theorem to guarantee the stability of the closedloop control system based on the prediction horizon length.Maurath et al. 98 proposed a minimum level for the prediction horizon based on the process delay and the control horizon (N C ), to obtain an invertible dynamic matrix with a full column rank.Similarly, Genceli et al. 99 defined a bound on the prediction horizon to assure the stability of the closed-loop control with constraints on outputs and inputs.They defined this bound based on the maximum ratio of the difference between the maximum and minimum of inputs over the maximum change of inputs.
Control horizon (N C ) denotes the number of time steps for which MPC calculates the optimal control action to minimize the specified cost function (N C ≤ N P ). 41Depending on the length of the control horizon, the control action can be aggressive or conservative, 31 which leads to a trade-off: raising the control horizon improves the robustness but may result in a more aggressive control action.On the other hand, a small control horizon (e.g., N c ¼ 1) degrades the robustness but improves the conservativeness solution of the problem and reduces the computational effort.Therefore, enhancing the performance of the closed-loop system by increasing the control horizon also increases the computational load.
Georgiou et al. 101 suggested to set the control horizon based on the time required for the plant to reach 60% of the steady state.Genceli et al. 99 purposed to set the control horizon based on the difference between control inputs to ensure robust stability of the controller.
Nagrath et al. 102 suggested a minimum value of the control horizon based on the number of unstable poles of the system.Similarly, Rawlings et al. 103 considered the state space representation of the model, Equation ( 1), and tuned the factors based on the poles of matrix A and the ability of the system to reach stability with A, B f g.Rossiter et al. 104 examined the impact of altering N C and concluded increasing it will improve the overall performance of the MPC if the prediction horizon (N P ) is long enough.Moreover, several researchers, including Maciejowski et al., 40 Seborg et al., 57 and others [105][106][107] implemented the input blocking concept to reduce the computational effort.They divided the time horizon required to reach the steady state into a set of blocks and calculated one control law for each block at each iteration.In this way, the computational complexity significantly decreases since the number of equations shrinks to only a specific set of blocks instead of an entire horizon.
Weighting Matrices in cost function (Q and R): In the trajectory tracking case, Equation (4), Q is employed to penalize any deflection from the trajectory, and R is used to minimize any abrupt control increment to result in a smooth transition.Large values of Q in comparison to R reflect the designer's intent to quickly drive the state to the preferred trajectory at the expense of large control action. 28reover, both matrices are usually diagonal with positive elements for weighting the most significant elements. 28,57For instance, if the effect of specific control action is to be decreased (matrix R), the associated diagonal element will be raised to highlight this intention.Similarly, if the prediction error at the end of the prediction horizon is more important than the beginning, then the related weight at the end of the horizon (matrix Q) ought to be increased to highlight its importance. 57Therefore, the choice of these matrix elements is typically left to the designer to adjust the control action based on the process requirements. 31mpling interval (T s ) is the period of time that the calculated control action remains constant.The sampling interval influences both the accuracy of the discretization and the resulting stability of the discrete-time model, as well as the controller's performance.However, T S should be small enough to reflect the effect of control action on the output and to reject the unknown disturbance. 38At each sampling time instant, the controller employs the system measurement and generates an optimization solution accordingly.Small T s yields a higher computational effort since recursive calculations must be processed in a shorter time.Accordingly, the choice of T S is driven by the computational capacity of the operating computer.As a general rule in control theory, a control system requires at least 10-20 samples during the rise time T 90 : 41 Authors in References 108-110 have used a sampling interval of 200 μs and less for their applications.
In summary, the goal is to finely discretize the time axis with the most minor sampling interval possible, considering the time required by the controller for computation, communication, and implementation of the control output to the system. 41

| Stability
Stability is the fundamental requirement for any control system, ensuring reliable and predictable performance even in challenging situations. 111,112The term 'stability' refers to the ability of a control system to respond within a specified range when subjected to bounded control inputs and unfavorable disturbances. 113Control theory offers a range of approaches to assess stability, such as evaluating the gains of linear and non-linear operators that describe the system's dynamic response. 114 the realm of control systems, uncertainties can jeopardize stability by pushing system states beyond the optimal region, hindering precise control. 4,28To mitigate this, it is crucial to ensure that the response of a controlled system remains bounded under all possible perturbations, both in nominal conditions and real-world applications.
While feedback control systems are commonly employed to achieve stability, it's important to be mindful of their potential to introduce instability through the injection of measurement noise into the controlled system. 1146][117] However, when employing MPC with constraints, conventional linear techniques or frequency-domain stability analysis methods are not directly applicable. 96One popular and straightforward approach in analyzing stability is the use of the Lyapunov stability method, which involves constructing a Lyapunov function within the cost function. 1113][124][125] Furthermore, recent work by Adinehvand et al. showcased the use of a time-varying Lyapunov function in motion control and trajectory tracking applications, effectively bounding trajectory errors. 126For a comprehensive overview of MPC stability theory, readers may refer to the excellent works by Rossiter et al. 104 and Mayne et al. 104,119 By prioritizing stability in control systems, particularly in the context of MPC, manufacturers can confidently navigate the complexities of bioprocessing and unleash the full potential of control strategies in their production processes.

| Robustness
A closed-loop system is robust if it is insensitive to variations in process dynamics.In other words, "When we say that a control system is robust we mean that stability is maintained and that the performance specifications are met for a specified range of model variations (uncertainty range)". 36,127Essentially, uncertainties, such as model discrepancies, can impede the controller from effectively guiding states to stable points. 28Therefore, it becomes essential to employ a robust controller that ensures stability, feasibility, and adherence to desired operation despite the uncertainties that may exist. 121 MPC control scheme inherently comes with a certain level of robustness against minor uncertainties due to its receding horizon nature. 128Although, it often falls short in systematically handling uncertainty due to its deterministic nature. 2 Consequently, additional methods and tools are needed to enhance the performance of the controlled system. 12,870][131][132][133][134] It is important to note that while most filtering techniques consider the variance of the measurement signal, the robustness of MPC primarily depends on bounding the prediction error. 135,136gnificant research efforts have been dedicated to improving the robustness of MPC.For instance, Magni et al. 137  studied and successfully implemented in academia and industry. 143 the other hand, Nonlinear MPC (NMPC) arises when either the model prediction or the cost function becomes nonlinear, significantly increasing the complexity of the optimization problem.
Hybrid MPC is another class of MPC that combines continuous and discrete forms of states and constraints within the controlled system.For instance, in the control of a thermostat system, the measured temperature value is continuous, while the heating/cooling device can only be on or off. 144Depending on the cost function's composition, it can be classified as mixed-integer linear programming (MILP) or mixed-integer quadratic programming (MIQP) if the cost function has a linear or quadratic form, respectively.When nonlinearity is incorporated, it is known as a mixed-integer nonlinear programming (MINLP) problem. 41An excellent explanation is presented by Sanfelice et al. 144 To address uncertainties, such as model discrepancy, various MPC schemes have been developed and employed.Robust MPC (RMPC), stochastic MPC (SMPC), and repetitive MPC are among these strategies, each offering specific approaches to tackle uncertainty and enhance control performance.These MPC schemes will be thoroughly discussed and studied in the following section.
It is worth mentioning that besides the MPC variants described here, other types of MPC have also been proposed and applied in different domains.These include distributed MPC, 145,146 economic MPC, 147,148 learning MPC, [149][150][151] and many more, [152][153][154][155] each tailored to specific control objectives and system characteristics.

| Robust MPC
Robust Model Predictive Control (RMPC) is a practical approach for ensuring satisfactory controller performance when dealing with realworld plants, where uncertainties and variations between the mathematical model and the actual plant behavior can significantly impact control system performance. 113This concept forms the foundation of robust MPC techniques, which aim to address uncertainties and disturbances that can impact control system performance.[158] RMPC enhances the traditional Model Predictive Control (MPC) framework by augmenting the mathematical model used for prediction and optimization with uncertainty information obtained during the modeling process.This incorporation of uncertainties allows the control system to adapt and respond robustly to variations between the model and the actual plant behavior, ensuring satisfactory controller performance in real-world applications.
Various methodologies have been developed within the framework of RMPC to address uncertainties, including tube-based methods, 159,160 min-max formulations, 73 and constraint-tightening methods. 3,161These approaches consider worst-case scenarios of model uncertainties within predefined boundaries, ensuring the control system remains robust and feasible even under challenging conditions.
While RMPC provides an optimization-based control synthesis approach that rigorously accounts for system constraints, performance requirements, and uncertainties, 29,121,161,162 it often yields conservative solutions by assuming worst-case scenarios and constraining control authority unnecessarily. 38This conservatism may lead to suboptimal outcomes in real-world applications where worstcase scenarios may not occur.
Nevertheless, RMPC offers computational tractability by simplifying objectives and uncertainties. 162It guarantees constraint satisfaction and robustness feasibility with low computational effort, even in the presence of model inaccuracies or uncertainties. 163,164Among the various RMPC approaches, the tube-based methodology has gained significant interest due to its straightforward approach to handling constraints. 157r readers interested in delving deeper into RMPC, we recommend referring to the following publications for in-depth details. 161,162,165 These resources provide valuable insights into the theoretical foundations, methodologies, and practical applications of RMPC.

| Stochastic model predictive control
This approach offers a valuable alternative to the deterministic approach presented in Equation (1) as it acknowledges the uncertainties inherent in real-world experiments resulting from measurement errors and potential model discrepancies.Unlike Robust MPC, which handles uncertainties deterministically with hard constraints, SMPC embraces the probabilistic nature of system parameters and model predictions. 156,166In addition, the SMPC technique can handle unbounded disturbances, distinguishing it from the RMPC scheme.
Conventionally, uncertainties are incorporated into the state space representation by adding them to the main equation (Equation ( 1)): where w k R nw denotes the model discrepancy and system disturbance, and v k R nv is measurement error.w k and v k are independent random variables with known covariance and probability distribution, where However, while the original system configuration is deterministic (Equation ( 1)), the introduction of random variables in the system dynamics (Equations ( 5), ( 6)) leads to a statistical representation of the system model.Consequently, the cost function and constraints need to be adapted to accommodate the probabilistic nature of model predictions.In SMPC, constraints are softened and treated probabilistically, allowing for violations with a given probability. 156These probabilistic constraints, referred to as chance constraints, are a fundamental feature of the SMPC scheme, providing a systematic trade-off between control performance and constraint violation probability. 41,167Mainly, there are two approaches for softening the constraints 161 ; one option is to employ the average of the hard constraint: Another option is to replace the hard constraints with the violating probability to reflect their uncertain nature: where E Hx ½ is the expected value of the constraints x Χ, and h j is the j-th element of the vector of h, and Pr k Hx ≤ h j Â Ã is the probability of satisfaction of constraint j for a given state at time step k.
defines the acceptable level of the constraint j-th.The goal is to ensure that the probability of violating any constraint (1 is less than or equal to α j for all constraints.
This methodology provides non-conservative optimization solutions by considering the most likely disturbances observed in practice, rather than relying on worst-case scenarios.Early work by Åström et al. 168 focused on optimizing a transfer function by minimizing variance within stochastic MPC.Schwarm et al. 169 presented the traditional formulation for handling probabilistic constraints, while Farina et al. 170,171 investigated the current approaches for solving the SMPC and classified their characteristics regarding the convergence properties, dynamic features, and probabilistic constraints.
SMPC has garnered attention in various fields, including traffic management, 172 robotics focused on path planning, 166 and regulating building climate control. 2,156The comprehensive research conducted by the authors in Reference 121 explores control and performance aspects of uncertain systems and proposes approaches to reduce computational costs in diverse applications.

| Repetitive model predictive control (ReMPC)
Once again, the overall performance of MPC is highly dependent on the quality of the mathematical model describing the process and the way of considering the future disturbance. 173,174Accordingly, the performance of an optimization problem with a low quality of any of these two aspects is likely to deteriorate.The ReMPC was proposed by Natarajan et al. 175 to enhance the MPC performance period-to-period for the purification process with simulated moving bed (SMB) chromatography.In fact, the authors augmented MPC by borrowing the periodic noise rejection concept from repetitive control (RC) to follow a repetitive reference trajectory. 176,177r applications that the system must follow a repetitive reference trajectory and involves a periodic disturbance, ReMPC provides superior performance compared to the classical MPC.Therefore, the essential idea of ReMPC is to integrate the effect of periodic disturbance from past periods to enhance the quality of future prediction.
However, the cost function must be designed in a way to be less sensitive to non-periodic disturbance.7][178][179][180][181][182][183] For instance, Erdem et al. 178 presented the automatic control of the SMB process by considering virtual eight-column SMB, to control the product purity at the raffinate and extract.They used the reduced-order model to utilize the RMPC around the cyclic-steady state.They employed the same assumptions and furthered their strategy by considering the separation under linear 179 and non-linear competitive isotherm adsorption, and generalized Langmuir isotherm. 179,180Refer to Sguarezi et al. 184 for further details on this strategy's implementation. 184

| APPLICATIONS
The effectiveness of MPC in optimizing control and decision-making under constraints has been demonstrated in various fields such as automated vehicles, aerial mechatronic systems, and autonomous marine vehicles.MPC offers a framework for identifying ideal operating conditions and deriving the process to these set points in the most feasible way while ensuring optimal control within constraints.The intuitive structuring of the control law as an optimization problem has made MPC appealing in many engineering domains.
The lessons learned from the advancements and implementations of MPC in other fields have significant potential for the biotechnology industry.6][187] Additionally, MPC can be used to optimize downstream processing, such as purification and separation, while maintaining product quality within the constraints of the process. 188,189The use of MPC in autonomous systems can be adapted to control and optimize complex bioprocesses, where high levels of automation and decision-making under constraints are required. 190erall, the success of MPC in various fields outside of biotechnology highlights its potential for application in bioprocess optimization and control.By learning from these advancements and implementing them in biotechnology, MPC can help improve process efficiency, product quality, and overall performance.

| Biotechnology
Production of biological products is often divided into two main steps, the upstream processing in a bioreactor and the downstream processing for the recovery of the product by a combination of centrifugation, filtration, chromatography, and membrane filtration.
As the upstream processing is the initial step in biotechnology, thus product quality is of paramount importance, and optimizing this step leads to significant improvements in overall production.
Upstream processing involves cell cultivation and harvesting biological material, such as proteins, enzymes, and antibodies, while maintaining its quality and purity.The outcome of this step is heavily dependent on controlling and maintaining process variables such as nutrition feeding, temperature, pH, and dissolved oxygen at an optimal level.This step can be performed via batch, fed-batch, or perfusion modes, each with its advantages and limitations.Similarly, downstream processing poses challenges due to the complexity and variability of the starting material.Multi-component adsorption isotherms, variable impurities, and contaminants in the crude feedstock make it challenging to design and optimize downstream processing steps, such as purification and separation.
In discussing the challenges faced by the biopharmaceutical industry in meeting the growing demand for biologics while maintaining quality requirements, Narayanan et al. 191 highlight the need for advancements in bioprocessing, particularly in process intensification, to address these challenges.In addition, they stress the importance of collaborations between different parties to establish a more digitalized and automated biotherapeutic process development and manufacturing, aligning the industry with the standards of Industry 4.0.
Model predictive control offers a promising solution to ensure the final product meets the required specifications.In addition, MPC can exploit the benefits of the process by minimizing by-products, improving productivity and yield, and even reducing production costs.
However, as stated before, the performance of any model-based controller, including MPC, largely depends on the accuracy of the process model used.In the context of bioprocessing, obtaining a reliable and accurate process model is particularly challenging given model uncertainties, the system's nonlinear structure, and the slow process dynamics. 192,193The complexity of process models primarily arises from the high sensitivity of biological products to process conditions, including temperature, pH, and substrate concentrations.In light of this, researchers have explored various strategies to overcome the challenge of obtaining a reliable model for utilizing MPC.These strategies include traditional approaches, such as model identification 194,195 and parameter estimation, 196,197 as well as more advanced techniques, including hybrid modeling approaches. 198,199ny authors have used and evaluated the use of MPC in the fed-batch and perfusion process.However, the results are mainly validated in simulations, raising the question of its applicability to real-world applications.There are numerous examples of MPC in simulation for cell culture process technology, [200][201][202][203][204][205] and few are implemented in practice.
Markana et al. 201 presented a novel control approach using economic MPC (EMPC) using an extended Kalman filter to locate the optimal conditions of a fed-batch reactor based on a multi-criteria objective.In addition, they used lexicographic optimization to identify the optimal substrate-feeding policy to maximize productivity and minimize the substrate rate simultaneously.The lexicographic optimization technique allowed for a prioritization of the objectives and a systematic approach that provided a Pareto-optimal solution, more information about lexicographic optimization, refer to the work by Gutjahr et al. 206 However, this optimization technique is a strict hierarchy optimization algorithm in terms of objective goals, and therefore the authors used a tolerance value for each goal to relax the constraints and avoid a local optimum (suboptimal results).In addition, the computational intensity of the lexicographic optimization technique may pose a challenge, especially for large-scale problems with many objectives.
A methodology for implementing MPC and moving horizon estimation (MHE) to optimize cultivation conditions for E. coli growth in parallel mini bioreactors is presented by Kim et al. 207 The proposed framework was used to control bolus feeding dynamically, and its effectiveness was demonstrated through in-silico experiments.This framework enabled the development of feed strategies that maximized process yield under various experimental conditions.Nevertheless, the outcome of the proposed control approach could be further improved and assessed by considering diverse objectives in the cost function, such as productivity, substrate utilization, and even minimization of by-product formation.Furthermore, despite the superior performance of MHE in compensating for the effect of model discrepancy or noisy measurement of the process, it might be computationally intractable, especially when it applies to multiple bioreactors simultaneously. 208An alternative is to employ a robust NMPC approach for regulating substrate flow rate in fed-batch culture to maximize biomass production while keeping acetate concentrations low. 205In addition, the authors evaluated the control system's robustness by employing a min-max optimization method to address the potential model mismatch's impact on the system.Nevertheless, the authors' validation of the results only involved Monte Carlo simulations, raising doubts about its effectiveness in real-world applications.
Kim et al. 209 employed a two-stage MPC framework for a fedbatch bioreactor and evaluated the proposed methodology through simulation studies.First, they used the reinforcement learning (RL) method as a supervisory controller to consider the lengthy cultivation time and to steer the process to the desired trajectory with specific productivity and yield; wherein in the inner control loop, they employed and compared the classic MPC and economy MPC for optimizing the overall performance of the system in the presence of model mismatch and real-time disturbance.In a subsequent work, 22,210 MPC was successfully implemented on a physical system.
The desired process yield was successfully achieved by the authors 210 through the implementation of setpoint tracking, using PAT technology.Rashedi et al. 22  that the drawbacks of MPC, including its perceived computational requirements, can be overcome.
In downstream processing, usually the longest step is purification at the chromatography step.It is worth mentioning that, as noted by Budzinski et al., 212 62% of buffer consumption in the entire process of producing monoclonal antibody (mAb) belongs to the chromatographic step.4][215][216][217][218][219][220][221][222] Suvarov et al. 215  It is true that using a reduced-order model can provide computational and practical advantages, but it must be noted that it may also introduce some trade-offs and challenges that must be carefully noted.Reduced-order models are approximations of the original system, which means that they may not capture all the necessary dynamics of the system.The primary disadvantage of utilizing reduced-order models for MPC is that it can be difficult to model the system dynamics precisely.The reduction in the number of states and inputs implies that the prediction accuracy of the model is also reduced, which can lead to suboptimal control performance and constraint violations if the reduced-order model is not sufficiently precise. 223,224Furthermore, reduced-order models can be more challenging to tune since model parameters must be adjusted to accommodate the reduced-order dynamics. 225Finally, the reducedorder model is more prone to disturbances, model reduction errors, and estimation errors, which can significantly impact the controller's performance.Developing and validating a reliable, accurate reducedorder model requires substantial expertise and computational resources. 226Therefore, it is crucial to carefully evaluate the advantages and limitations of using a reduced-order model for MPC, considering the specific complex system being controlled. 227lami et al. 132 employed MPC in parallel with an extended Kalman filter for optimizing the process conditions during the loading phase, Figure 4.The optimization goal in their work was to maximize productivity and resin utilization and optimize the buffer consumption simultaneously.In addition, they used a simple mechanistic model, the linear driving force (LDF), based on pore diffusion adsorption to avoid the extra computational burden and managed to compensate for the effect of model discrepancy via the proposed model structure.As a result, the proposed MPC was experimentally validated with both affinity and cation exchange chromatography, demonstrating its applicability to real-world applications.Subsequently, they were successfully able to maintain the resin utilization at the highest level but increased the productivity beyond the feasible region in batch chromatography.Hence, we managed to effectively reduce the processing time by half while maintaining a certain level of resin utilization, leading to a significant decrease in buffer consumption.Furthermore, the proposed system would also be ideal for compensating for the impact of column aging, but a soft sensor for the feed stream product would be a prerequisite.Fortunately, the viability of such sensors has been demonstrated through previous successes. 228,229 crystallization, agglomeration and unwanted fines are two problems often encountered in industrial crystallization processes, which may result in more prolonged drying and filtration. 231To this aim, several studies, including those by Rathore et al., 190 Szilagyi et al., 231 and others [232][233][234][235][236][237][238] have highlighted the benefits of optimizing with MPC in crystallizing active pharmaceutical ingredients (API).A popular technique involves optimizing the crystallizer cooling profile to control the crystal size distribution. 234,236,239Sen et al. 233 employed a hybrid MPC-PID control strategy, in-silico, for continuous purification of APIs, including crystallization, filtration, and drying with a tablet manufacturing unit operation.They demonstrated that the hybrid MPC-PID control scheme outperformed a PID-only control scheme, showcasing its potential to enhance the efficiency of pharmaceutical manufacturing operations.
MPC has also been implemented in feeding blending unit operation to follow a reference trajectory, with a constant mass flow at constant concentration, by adjusting the blender speed.

| CONCLUSIONS
In the past two decades, Model Predictive Control has emerged as a widely used control tool in industry and academia.Its popularity stems from its simplicity in development, direct physical understanding of parameters, and effective handling of constraints, provided a suitable model is available. 36This study has investigated and consolidated various MPC implementations for optimizing processes in the presence of model discrepancies and measurement inaccuracies.In addition, it has examined essential design factors related to controller robustness, closed-loop system stability, and the formulation of the underlying optimization problem.
Based on the evaluation of popular MPC techniques, it can be concluded that accurately developed and adapted MPC for specific case studies can lead to improved system performance. 87The main hurdle to overcome is the complexity of modeling and tuning the designing parameters, such as prediction/control horizons and sampling interval.Achieving optimal performance heavily relies on the quality of the process model.Striking a balance between model precision and computational complexity is crucial, as overly complex models can hinder computational tractability.[251] The cost function is a vital part of the MPC.A proper design of the cost function guarantees the stability of the optimization.At the same time, the tuning parameters will improve the overall outcome of the process.For instance, in systems with fast dynamics, the timing cycle must be as small as possible to achieve a smooth outcome with fine granularity of control output.

F I G U R E 4
The structure of the control strategy incorporated MPC for optimization and EKF for state estimation, adopted from Eslami et al. 230 y k represents the real-time measurement, u k represents the computed control action, and x k is the optimized state estimation.
As we cast our gaze forward, it's hard to ignore the burgeoning promise of MPC within the realm of Integrated Continuous Bioprocessing (ICB).The biopharmaceutical world is steadily leaning towards ICB, and in this shift, MPC stands out as a beacon of advanced control.Imagine the continuous crafting of monoclonal antibodies, or the melding of MPC with the ever-evolving world of artificial intelligence.And when we think of MPC intertwining with hybrid modeling, it paints a picture of a future where control and predictability in biomanufacturing reach unprecedented heights.We anticipate that the global shift towards sustainable processes will drive the increased utilization of MPC in bioprocess units, enabling the handling of sluggish systems with inaccurate forecasting models.
The integration of model-based predictive controllers into various applications, including continuous upstream and downstream processing, holds great potential.Based on our observations, we expect a significant growth in the number of MPC applications due to existing disciplines and emerging trends.It is our hope that this review article will inspire readers to implement and further explore this intelligent control approach, fostering advancements in manufacturing processes.
these factors are biased toward each other.Hence the goal is to locate a good compromise between the objectives.Another advantage of using multi-objective MPC is that the user can anticipate the outcome based on the cumulative performance goals and adjust the objectives or their weights to tailor the control strategy to the specific task at hand.It is worth mentioning that different mathematical formulations of the objective function might result in varied MPC problem classes with diverse complexity levels and computing difficulty.Accordingly, the cost function structure directly influences the choice of appropriate control strategy (e.g., linear/non-linear MPC).4,41In this article, we have explained the following optimization techniques to cover the typical applications of the MPC controller in academia and industry: trajectory tracking, goal attainer, minimax, and Pareto front.
used a novel artificial neural network (ANN) based approach for tuning the weighting factors in real time.They demonstrated that the performance of the controller improves when the weighting factors are adaptively adjusted based on the variable reference in the cost function.
conducted a study using a hybrid model-based MPC approach to maximize cell growth and metabolite production in fed-batch bioprocesses.By integrating the hybrid model into the MPC framework, the researchers aimed to enhance the overall performance and efficiency of the bioprocess and successfully improved the final production rate by 2%.In a related study, 211 various control methods for substrate feed rate in bioprocesses, including MPC, artificial neural networks (ANNs), and fuzzy logic controllers, were discussed.The MPC system exhibited superior control performance and reduced process variability compared to open-loop feeding profile control.The authors demonstrate a practical example of implementing a robust and flexible MPC system on standard computer hardware, indicating have presented a simple robust MPC for simulated moving bed (SMB) chromatography with four columns based on a simple mechanistic model, namely linear driving force.They evaluated the performance of the proposed control strategy's performance in controlling the product's purity at the outlet with multiple numerical simulations.Grossmann et al.214 proposed an optimizing model predictive control algorithm for the chromatographic multi-column solvent gradient purification (MCSGP) process, which was demonstrated through two case studies.The first case study focused on separating three antibody variants, while the second involved the separation of a peptide mixture within a 3-column MCSGP.By using a lumped kinetic model to anticipate the system dynamics, the authors proved the controller's effectiveness in meeting product specifications and maximizing productivity.However, the computational expense of the model led the authors to simplify and linearize it around the steady-state to obtain a reduced order model (ROM) at each iteration.To account for model uncertainties, they applied a Kalman filter.Overall, the proposed approach maintained the purity of the target product at a high level while maximizing yield.However, the results were limited to simulated experiments since real-world implementation would require accurate measurements, which may not be available in systems without online monitoring.Papathanasiou et al.182,216 proposed a PARametric Optimization and Control Framework (PAROC) to develop and evaluate the multiparametric model predictive control (mp-MPC).The proposed framework consists of four steps, (1) developing a high-fidelity model, (2) reducing the model order, (3) designing a multiparametric controller, (4) validating the performance of the obtained controller regarding constraint satisfaction, stability, and set point tracking via multiple simulations.
240 A low-pass finite impulse response (FIR) to filter disturbance from the online data was used, and the performance of the MPC controller compared with three other scenarios involving a classic PI controller.The results highlighted the effectiveness of MPC in achieving precise trajectory tracking and its superiority over alternative control strategies.Kirchengast et al.241 employed MPC combined with a Smith predictor based on a data-driven linear model in direct compaction in continuous tablet production.They proposed this strategy to guarantee the API concentration and discard the out-of-specification material before it enters the tablet press hopper level.A hybrid MPC-PID control strategy is implemented in References 242,243 to enhance the quality of direct compaction tablet manufacturing.The authors used the MPC based on real-time monitoring tools and principal components analysis (PCA) as a supervisory level in a pilot plant.In a further study, Singh et al.244,245 investigated several feedback and feedforward control architectures, including MPC, to identify the optimal solution for continuous tablet manufacturing via direct compaction, roller compaction, and wet granulation manufacturing routes.To bridge the gap between upstream processing and downstream processing in the context of continuous pharmaceutical manufacturing, researchers have explored the implementation of MPC in an endto-end approach.Mesbah et al.246 assessed MPC with a non-linear plant-wide simulator for end-to-end continuous pharmaceutical manufacturing, starting from synthesis step and reactors until the tableting step.Accordingly, they considered the proposed control strategy to facilitate products' critical quality attributes (CQAs) in the presence of uncertainties and possible drifts in operating units.In,247 they advanced by utilizing stochastic MPC for end-to-end continuous pharmaceutical manufacturing and considering uncertainties by a probability density function.Sacher et al.248 implemented MPC to synthesize drug substances continuously and validated the results experimentally.They employed various model types for each phase of the synthesis process; they used a mechanistic model based on the reaction kinetics and data-driven using online data based on the local linear model tree (LOLIMOT) to maximize yield and minimize side products.
Linear MPC (LMPC) refers to MPC formulations where both the model prediction and cost function have a linear or quadratic structure.LMPC can be considered a mature control technique, extensively 142mulated a robust controller by employing a quadratic cost function with a nonlinear system model, enabling it to handle bounded disturbances and parameter uncertainties.Kellett et al.138demonstrated that a control system is inherently robustly stable if it exhibits a continuous Lyapunov function.Yu et al.128investigated the robustness of continuous-time nonlinear MPC with input and terminal constraints.Liu et al.139proposed a technique to achieve robust MPC by utilizing a non-square cost function without imposing additional stability constraints.Model Predictive Control has evolved from its origins in the chemical process industry during the late 1970s.142Initiallydesigned for systems with linear dynamics, researchers progressively extended MPC to optimize processes under various conditions.It is important to note that besides the MPC types discussed below, there are other variants of MPC that have been developed and utilized in different applications.