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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

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Dynamic optimization plays an important role toward improving the operation of chemical systems, such as batch and semi-batch processes. The preferred strategy to solve constrained nonlinear dynamic optimization problems is to use a so-called direct approach. Nevertheless, based on the problem at hand and the solution algorithm used, direct approaches may lead to large computational times. Indirect approaches based on Pontryagin’s Minimum Principle (PMP) represent an efficient alternative for the optimization of batch and semi-batch processes. This paper, which is an extension to our ESCAPE-2017 contribution 2, 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. In addition, this article discusses structural differences between direct and indirect approaches. The proposed method is tested on both 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 method and a direct simultaneous method. It is observed that the combination of the indirect approach with parsimonious input parameterization can lead to significant reduction in computational time.

The trend towards high-quality, low-volume chemical production has put more emphasis on batch and semi-batch processing due to its increased operational flexibility. The transient behavior of these processes makes their real-time optimization very challenging. In particular, the large prediction horizons required in shrinking-horizon NMPC increase the real-time computational effort due to expensive matrix factorizations. 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. This study proposes to combine a tailored parsimonious input parameterization with shrinking-horizon NMPC to reduce the real-time computational effort. Models of the optimal solution are used to suggest parsimonious parameterizations (especially for sensitivity-seeking arcs) that lead to computationally efficient optimization. The proposed approach is illustrated in simulation on two case studies in the presence of uncertainty, namely a batch binary distillation column and a semi-batch reactor for the hydroformylation of 1-dodecene. The results show that the tailored parsimonious shrinking-horizon NMPC (i) performs very similarly to the standard shrinking-horizon NMPC in terms of cost, (ii) is computationally much more efficient than the standard shrinking-horizon NMPC especially at the beginning of the batch, (iii) is robust to plant-model mismatch.

A batch process is characterized by the repetition of time-varying operations of finite duration. Due to the repetition, there are two independent "time" variables, namely, the run time during a batch and the batch index. Accordingly, the control and optimization objectives can be defined for a given batch or over several batches. This chapter describes the various control and optimization strategies available for the operation of batch processes. These include online and run-to-run control on the one hand, and repeated numerical optimization and optimizing control on the other. Several case studies are presented to illustrate the various approaches.

Nonlinear model predictive control (NMPC) is an important tool for the real-time optimization of batch and semi-batch processes. Direct methods are often the methods of choice to solve the corresponding optimal control problems, in particular for large-scale problems. However, the matrix factorizations associated with large prediction horizons can be computationally demanding. In contrast, indirect methods can be competitive for smaller-scale problems. Furthermore, the interplay between states and co-states in the context of Pontryagin’s Minimum Principle (PMP) might turn out to be computationally quite efficient.
This work proposes to use an indirect solution technique in the context of shrinking-horizon NMPC. In particular, the technique deals with path constraints via indirect adjoining, which allows meeting active path constraints explicitly at each iteration. Uncertainties are 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.
Keywords: Nonlinear model predictive control, indirect optimization methods, indirect adjoining, semi-batch processes, Pontryagin’s Minimum Principle

We present a sensitivity-based predictor-corrector path-following algorithm for fast nonlinear model predictive control (NMPC) and demonstrate it on a large case study with an economic cost function. The path-following method is applied within the advanced-step NMPC framework to obtain fast and accurate approximate solutions of the NMPC problem. In our approach, we solve a sequence of quadratic programs to trace the optimal NMPC solution along a parameter change. A distinguishing feature of the path-following algorithm in this paper is that the strongly-active inequality constraints are included as equality constraints in the quadratic programs, while the weakly-active constraints are left as inequalities. This leads to close tracking of the optimal solution. The approach is applied to an economic NMPC case study consisting of a process with a reactor, a distillation column and a recycler. We compare the path-following NMPC solution with an ideal NMPC solution, which is obtained by solving the full nonlinear programming problem. Our simulations show that the proposed algorithm effectively traces the exact solution.

This work considers the numerical optimization of constrained batch and semi-batch processes, for which direct as well as indirect methods exist. Direct methods are often the methods of choice, but they exhibit certain limitations related to the compromise between feasibility and computational burden. Indirect methods, such as Pontryagin’s Minimum Principle (PMP), reformulate the optimization problem. The main 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.
This study presents an alternative 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.
Keywords: Constrained dynamic optimization, Pontryagin’s Minimum Principle, indirect optimization methods, quasi-Newton algorithm, semi-batch processes

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.

Due to hardening competition and increased focus on resource efficiency, efforts are made to develop advanced industrial optimization and control systems with the goal to shift the (semi-)batch production from recipe-based to a state-based approach. This study illustrates the steps needed for the implementation of optimization and on-line control of semibatch emulsion copolymerization involving the development of process model, its validation and connection with control software, and the realization at pilot plant scale. The process model must be fast and robust enough to provide estimation of the process trajectory reliably and quickly. Moreover, in connection with nonlinear model predictive control (NMPC), the model has to be able to learn from the process and to update parameter values in real time, e.g., due to change of reactor jacket heat transfer. The Cybernetica CENIT software is employed for NMPC. The industrial pilot-scale semibatch emulsion copolymerization of four comonomers (two of them water soluble) is used for the demonstration of NMPC functionality for: (i) reactor temperature control, (ii) minimization of batch time while preserving product quality, and (iii) minimization of batch duration with desired simultaneous shift in product quality.

Optimal processes often exhibit active path constraints. Parametric uncertainties in the process model might thus lead to constraint violations. A heuristic approach is presented to overcome this challenge. The nominal model is optimized with additional path constraints due to worst-case models. A heuristic method of choosing these models is proposed based on sensitivities of the constraints with respect to the uncertain parameters. The presented approximation does not guarantee robust feasibility, but path constraint violations are less likely to occur compared to the optimization using the nominal model solely. Two case studies are presented: a complex emulsion copolymerization process (DAE with 139 equations) and the penicillin formation (four differential equations and two algebraic equations). The results of both case studies show that, in contrast to the optimization in the nominal case, the multi-model approach does not violate the path constraints for different scenarios of the parametric uncertainty set.

Model predictive control (MPC) has demonstrated exceptional success for the high-performance control of complex systems [1], [2]. The conceptual simplicity of MPC as well as its ability to effectively cope with the complex dynamics of systems with multiple inputs and outputs, input and state/output constraints, and conflicting control objectives have made it an attractive multivariable constrained control approach [1]. MPC (a.k.a. receding-horizon control) solves an open-loop constrained optimal control problem (OCP) repeatedly in a receding-horizon manner [3]. The OCP is solved over a finite sequence of control actions {u<sub>0</sub>,u<sub>1</sub>,f,u<sub>N- 1</sub>} at every sampling time instant that the current state of the system is measured. The first element of the sequence of optimal control actions is applied to the system, and the computations are then repeated at the next sampling time. Thus, MPC replaces a feedback control law π(·), which can have formidable offline computation, with the repeated solution of an open-loop OCP [2]. In fact, repeated solution of the OCP confers an "implicit" feedback action to MPC to cope with system uncertainties and disturbances. Alternatively, explicit MPC approaches circumvent the need to solve an OCP online by deriving relationships for the optimal control actions in terms of an "explicit" function of the state and reference vectors. However, explicit MPC is not typically intended to replace standard MPC but, rather, to extend its area of application [4]-[6].