Optimal control in chemical engineering: Past, present and future

Abstract Roger W.H. Sargent (1926–2018) was an unprecedented pioneer who foresaw the role that mathematical and computational tools would have in chemical engineering. His visionary work created the multidisciplinary field of Process Systems Engineering (PSE), a field that acts as a central hub influencing all subfields of chemical engineering. His particular interest in optimal control applied to industrial processes led him to develop numerical techniques to solve large-scale optimal control problems. In this work, a brief overview of the theory of optimal control is offered, spanning from its roots in calculus of variations to Pontryagin’s maximum principle and some of its extensions. Furthermore, important contributions made by Sargent and his students are presented. Selected applications currently found in literature are presented as well–ranging from classical chemical engineering systems to bioprocesses. Some future perspectives of the field are also presented in the concluding section.

[1]  Efstratios N. Pistikopoulos,et al.  Towards an efficient numerical procedure for mixed integer optimal control , 1997 .

[2]  Lorenz T. Biegler,et al.  Dynamic optimization of HIPS open-loop unstable polymerization reactors , 2005 .

[3]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[4]  Rui Nian,et al.  A review On reinforcement learning: Introduction and applications in industrial process control , 2020, Comput. Chem. Eng..

[5]  Vassilios S. Vassiliadis,et al.  Stability criterion for the intensification of batch processes with model predictive control , 2018, Chemical Engineering Research and Design.

[6]  Lorenz T. Biegler,et al.  An indirect approach for singular optimal control problems , 2020, Comput. Chem. Eng..

[7]  Lars Imsland,et al.  Output feedback stochastic nonlinear model predictive control for batch processes , 2019, Comput. Chem. Eng..

[8]  Y. Ho.  Differential games, dynamic optimization, and generalized control theory , 1970 .

[9]  Paul I. Barton,et al.  DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis , 2014, BMC Bioinformatics.

[10]  K. Choi,et al.  On-Line Parameter Estimation in a Continuous Polymerization Process , 1996 .

[11]  Leonida Tonelli,et al.  Sur une méthode directe du calcul des variations , 1915 .

[12]  Olaf Stursberg,et al.  Applied Hybrid System Optimization: An Empirical Investigation of Complexity ? , 2004 .

[13]  Ignacio E. Grossmann,et al.  Simultaneous Cyclic Scheduling and Control of a Multiproduct CSTR Reactor , 2006 .

[14]  L. Biegler,et al.  Fast economic model predictive control based on NLP-sensitivities , 2014 .

[15]  Adam L. Meadows,et al.  Application of dynamic flux balance analysis to an industrial Escherichia coli fermentation. , 2010, Metabolic engineering.

[16]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[17]  Fahad Albalawi,et al.  Process operational safety via model predictive control: Recent results and future research directions , 2017, Comput. Chem. Eng..

[18]  Paul I. Barton,et al.  Nonsmooth differential-algebraic equations in chemical engineering , 2017, Comput. Chem. Eng..

[19]  Paul I. Barton,et al.  A vector forward mode of automatic differentiation for generalized derivative evaluation , 2015, Optim. Methods Softw..

[20]  Paul I. Barton,et al.  Mixed-integer dynamic optimization , 1997 .

[21]  Paul I. Barton,et al.  Global Optimization with Nonlinear Ordinary Differential Equations , 2006, J. Glob. Optim..

[22]  A. A. Patwardhan,et al.  NONLINEAR MODEL PREDICTIVE CONTROL , 1990 .

[23]  Basil Kouvaritakis,et al.  Striped Parameterized Tube Model Predictive Control , 2016, Autom..

[24]  M. Hestenes.  Calculus of variations and optimal control theory , 1966 .

[25]  Richard B. Vinter,et al.  Optimal Control Problems with Mixed and Pure State Constraints , 2016, SIAM J. Control. Optim..

[26]  Sigurd Skogestad,et al.  Data-driven Scenario Selection for Multistage Robust Model Predictive Control , 2018 .

[27]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[28]  J. Heijnen,et al.  Quantitative physiological study of the fast dynamics in the intracellular pH of Saccharomyces cerevisiae in response to glucose and ethanol pulses. , 2008, Metabolic engineering.

[29]  M. A. Henson,et al.  Genome‐scale analysis of Saccharomyces cerevisiae metabolism and ethanol production in fed‐batch culture , 2007, Biotechnology and bioengineering.

[30]  Ken R. Morison,et al.  Optimization of multistage processes described by differential-algebraic equations , 1986 .

[31]  Efstratios N. Pistikopoulos,et al.  New algorithms for mixed-integer dynamic optimization , 2003, Comput. Chem. Eng..

[32]  R. Sargent.  Optimal control , 2000 .

[33]  Richard B. Vinter,et al.  Optimal multiprocesses , 1989 .

[34]  Numerical Nonsmooth Optimization , 2020 .

[35]  Vassilios S. Vassiliadis,et al.  Simulation and optimization of dynamic flux balance analysis models using an interior point method reformulation , 2018, Comput. Chem. Eng..

[36]  Claire S. Adjiman,et al.  A Rigorous Global Optimization Algorithm for Problems with Ordinary Differential Equations , 2002, J. Glob. Optim..

[37]  David Q. Mayne,et al.  Model predictive control: Recent developments and future promise , 2014, Autom..

[38]  Hong Jang,et al.  A robust NMPC scheme for semi-batch polymerization reactors , 2016 .

[39]  W. L. Araújo,et al.  Microbial interactions: ecology in a molecular perspective , 2016, Brazilian journal of microbiology : [publication of the Brazilian Society for Microbiology].

[40]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[41]  Lorenz T. Biegler,et al.  Improved infeasible path optimization for sequential modular simulators. , 1983 .

[42]  L. Biegler,et al.  Robust stability of economically oriented infinite horizon NMPC that include cyclic processes , 2012 .

[43]  Walter Kähm,et al.  Lyapunov exponents with Model Predictive Control for exothermic batch reactors , 2018 .

[44]  Paul I. Barton,et al.  Computationally relevant generalized derivatives: theory, evaluation and applications , 2018, Optim. Methods Softw..

[45]  M. Tomita,et al.  Metabolic flux analysis and visualization. , 2011, Journal of proteome research.

[46]  J. A. Bryson.  Optimal control-1950 to 1985 , 1996 .

[47]  Matthias Gerdts,et al.  Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two , 2006 .

[48]  Carlos Eduardo García Sánchez,et al.  Comparison and analysis of objective functions in flux balance analysis. , 2014 .

[49]  Amir Beck,et al.  First-Order Methods in Optimization , 2017 .

[50]  Vassilios S. Vassiliadis,et al.  Thermal stability criterion integrated in model predictive control for batch reactors , 2018, Chemical Engineering Science.

[51]  F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control , 2013 .

[52]  Vassilios S. Vassiliadis,et al.  Optimal Laypunov exponent parameters for stability analysis of batch reactors with Model Predictive Control , 2018, Comput. Chem. Eng..

[53]  Dominique Bonvin,et al.  Dynamic optimization of constrained semi-batch processes using Pontryagin's minimum principle - An effective quasi-Newton approach , 2017, Comput. Chem. Eng..

[54]  J E Bailey,et al.  Application of mathematical tools for metabolic design of microbial ethanol production. , 1998, Biotechnology and bioengineering.

[55]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[56]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[57]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[58]  P I Barton,et al.  A reliable simulator for dynamic flux balance analysis , 2013, Biotechnology and bioengineering.

[59]  Timothy J. Hanly,et al.  Dynamic flux balance analysis for synthetic microbial communities. , 2014, IET systems biology.

[60]  R. B. Vinter,et al.  Existence of optimal controls for continuous time infinite horizon problems , 1997 .

[61]  L. Biegler,et al.  Large-scale decomposition for successive quadratic programming , 1988 .

[62]  Warren B. Powell,et al.  Approximate Dynamic Programming - Solving the Curses of Dimensionality , 2007 .

[63]  D. Kompala,et al.  Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures. , 1999, Journal of biotechnology.

[64]  Rishi Jain,et al.  Bayesian-based selection of metabolic objective functions , 2007 .

[65]  Bojan Likar,et al.  Predictive control of a gas-liquid separation plant based on a Gaussian process model , 2007, Comput. Chem. Eng..

[66]  Richard B. Vinter,et al.  Optimal Control , 2000 .

[67]  Francis Clarke,et al.  Optimal Control Problems with Mixed Constraints , 2010, SIAM J. Control. Optim..

[68]  Héctor J. Sussmann.  A Bang-Bang Theorem with Bounds on the Number of Switchings , 1979 .

[69]  Miguel J. Bagajewicz,et al.  On a New MILP Model for the Planning of Heat-Exchanger Network Cleaning† , 2004 .

[70]  Denis Dochain,et al.  State and parameter estimation in chemical and biochemical processes: a tutorial , 2003 .

[71]  Vassilios S. Vassiliadis,et al.  Optimisation of heat exchanger network cleaning schedules: Incorporating uncertainty in fouling and cleaning model parameters , 2019, Comput. Chem. Eng..

[72]  Erwin P. Gianchandani,et al.  The application of flux balance analysis in systems biology , 2010, Wiley interdisciplinary reviews. Systems biology and medicine.

[73]  Benoît Chachuat,et al.  Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control , 2014, J. Optim. Theory Appl..

[74]  Rainer Manuel Schaich,et al.  Robust model predictive control , 2017 .

[75]  Sebastian Engell,et al.  Efficient Robust Economic Nonlinear Model Predictive Control of an Industrial Batch Reactor , 2014 .

[76]  Daniel Sarabia,et al.  Improving scenario decomposition algorithms for robust nonlinear model predictive control , 2015, Comput. Chem. Eng..

[77]  Radoslav Paulen,et al.  Dual robust nonlinear model predictive control: A multi-stage approach , 2018, Journal of Process Control.

[78]  David Banjerdpongchai,et al.  Comparison of PID control and linear model predictive control application to regenerative thermal oxidizer system , 2015, 2015 10th Asian Control Conference (ASCC).

[79]  M. L. Shuler,et al.  Structured model for Saccharomyces cerevisiae , 1989 .

[80]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[81]  H. Sussmann,et al.  A maximum principle for hybrid optimal control problems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[82]  D. Mayne,et al.  Min-max feedback model predictive control for constrained linear systems , 1998, IEEE Trans. Autom. Control..

[83]  J. Frédéric Bonnans,et al.  A Shooting Algorithm for Optimal Control Problems with Singular Arcs , 2012, J. Optim. Theory Appl..

[84]  F. Doyle,et al.  Dynamic flux balance analysis of diauxic growth in Escherichia coli. , 2002, Biophysical journal.

[85]  David Q. Mayne,et al.  Robust model predictive control of constrained linear systems with bounded disturbances , 2005, Autom..

[86]  Dominique Bonvin,et al.  NMPC using Pontryagin's Minimum Principle-Application to a two-phase semi-batch hydroformylation reactor under uncertainty , 2018, Comput. Chem. Eng..

[87]  Sirish L. Shah,et al.  Quantifying the impact of model-plant mismatch on controller performance , 2010 .

[88]  Hans Müller-Steinhagen,et al.  Heat Exchanger Fouling in the Pre-Heat Train od a Crude Oil Distillation Unit - The Development of a "User Guide" , 2002 .

[89]  J. Heijnen,et al.  In vivo kinetics with rapid perturbation experiments in Saccharomyces cerevisiae using a second-generation BioScope. , 2006, Metabolic engineering.

[90]  U. Sauer,et al.  Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli , 2007, Molecular systems biology.

[91]  J. Doyle,et al.  Essentials of Robust Control , 1997 .

[92]  R.W.H. Sargent,et al.  Off Line Computation of Optimum Controls for a Plate Distillation Column* Calcul en dehors du circuit des commandes optimales pour une colonne de distillation h plateaux Off-line-Berechnung optimaler Regelungen f'tir den Boden einer Destillationskolonne , 1970 .

[93]  Lorenz T. Biegler,et al.  Integrated Optimization Strategies for Dynamic Process Operations , 2017, Theoretical Foundations of Chemical Engineering.

[94]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[95]  Timothy J. Hanly,et al.  Dynamic model‐based analysis of furfural and HMF detoxification by pure and mixed batch cultures of S. cerevisiae and S. stipitis , 2014, Biotechnology and bioengineering.

[96]  M. Petit.  Dynamic optimization. The calculus of variations and optimal control in economics and management : by Morton I. Kamien and Nancy L. Schwartz. Second Edition. North-Holland (Advanced Textbooks in Economics), Amsterdam and New York, 1991. Pp. xvii+377. ISBN0-444- 01609-0 , 1994 .

[97]  Nikolaos V. Sahinidis,et al.  A branch-and-reduce approach to global optimization , 1996, J. Glob. Optim..

[98]  Victor M. Zavala,et al.  Robustly stable economic NMPC for non-dissipative stage costs , 2017 .

[99]  Radoslav Paulen,et al.  Robust Nonlinear Model Predictive Control with Reduction of Uncertainty Via Robust Optimal Experiment Design , 2014 .

[100]  L. Biegler,et al.  Dynamic Optimization in the Design and Scheduling of Multiproduct Batch Plants , 1996 .

[101]  Stephen P. Boyd,et al.  Proximal Algorithms , 2013, Found. Trends Optim..

[102]  Artur M. Schweidtmann,et al.  Efficient multiobjective optimization employing Gaussian processes, spectral sampling and a genetic algorithm , 2018, Journal of Global Optimization.

[103]  Yurii Nesterov,et al.  Lexicographic differentiation of nonsmooth functions , 2005, Math. Program..

[104]  Lazaros G. Papageorgiou,et al.  Optimal Energy and Cleaning Management in Heat Exchanger Networks Under Fouling , 2000 .

[105]  R. Bellman.  Dynamic programming. , 1957, Science.

[106]  Min-Sen Chiu,et al.  Robust PID controller design via LMI approach , 2002 .

[107]  Lorenz T. Biegler,et al.  Advanced optimization strategies for integrated dynamic process operations , 2017, Comput. Chem. Eng..

[108]  Franziska Wulf,et al.  Minimization Methods For Non Differentiable Functions , 2016 .

[109]  Ignacio E. Grossmann,et al.  A tribute to professor roger sargent: Intellectual leader of process systems engineering , 2016 .

[110]  Lorenz T. Biegler,et al.  Nonlinear Waves in Integrable and Nonintegrable Systems , 2018 .

[111]  J. Kocijan,et al.  Gaussian process model based predictive control , 2004, Proceedings of the 2004 American Control Conference.

[112]  Zhe Wu,et al.  On integration of feedback control and safety systems: Analyzing two chemical process applications , 2018 .

[113]  B. Mordukhovich.  Variational Analysis and Generalized Differentiation II: Applications , 2006 .

[114]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[115]  Athanassios Sambanis,et al.  Modeling of cell culture processes , 2004, Cytotechnology.

[116]  Jean-Yves Audibert.  Optimization for Machine Learning , 1995 .

[117]  Paul I. Barton,et al.  Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations , 2012, Journal of Optimization Theory and Applications.

[118]  F. Clarke.  Necessary Conditions In Dynamic Optimization , 2005 .

[119]  Nagasuma R. Chandra,et al.  Flux balance analysis of biological systems: applications and challenges , 2009, Briefings Bioinform..

[120]  Mingzhao Yu,et al.  A reduced regularization strategy for economic NMPC , 2019 .

[121]  Maria do Rosário de Pinho,et al.  On Optimal Control Problems with Nonregular Mixed Constraints , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[122]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[123]  D. Himmelblau,et al.  Optimal control via collocation and non-linear programming , 1975 .

[124]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[125]  Sebastian Engell,et al.  Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty , 2013 .

[126]  Gilbert Ames Bliss.  The Problem of Lagrange in the Calculus of Variations , 1930 .

[127]  Fahad Albalawi,et al.  A feedback control framework for safe and economically‐optimal operation of nonlinear processes , 2016 .

[128]  R. Findeisen,et al.  Fully Parameterized Tube MPC , 2011 .

[129]  Moritz Diehl,et al.  The integer approximation error in mixed-integer optimal control , 2012, Math. Program..

[130]  Dongda Zhang,et al.  Reinforcement Learning for Batch Bioprocess Optimization , 2019, Comput. Chem. Eng..

[131]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[132]  Constantinos C. Pantelides,et al.  Optimal Control of Systems Described by Index Two Differential-Algebraic Equations , 1995, SIAM J. Sci. Comput..

[133]  Ming Yuchi,et al.  Online parameter estimation for MPC model uncertainties based on LSCR approach , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[134]  J. E. Cuthrell,et al.  Simultaneous optimization and solution methods for batch reactor control profiles , 1989 .

[135]  L. Biegler,et al.  A Stable and Robust NMPC Strategy with Reduced Models and Nonuniform Grids , 2016 .

[136]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[137]  Emil Karlsen.  A method for dynamic flux balance analysis with global constraints , 2017 .

[138]  Alberto Bemporad,et al.  Scenario-based model predictive control of stochastic constrained linear systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[139]  Benedetto Piccoli,et al.  Hybrid systems and optimal control , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[140]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[141]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[142]  Lorenz T. Biegler,et al.  Reduced successive quadratic programming implementation for large-scale optimization problems with smaller degrees of freedom , 1990 .

[143]  Lorenz T. Biegler,et al.  Improved infeasible path optimization for sequential modular simulators—II: the optimization algorithm , 1985 .

[144]  Christodoulos A. Floudas,et al.  A review of recent advances in global optimization , 2009, J. Glob. Optim..

[145]  Vassilios S. Vassiliadis,et al.  Mitigation of fouling in refinery heat exchanger networks by optimal management of cleaning , 2001 .

[146]  Andreas Potschka,et al.  Partial stability for nonlinear model predictive control , 2017, Autom..

[147]  L. Young,et al.  Lectures on the Calculus of Variations and Optimal Control Theory. , 1971 .

[148]  P. I. Barton,et al.  Global methods for dynamic optimization and mixed-integer dynamic optimization , 2006 .

[149]  E. J. McShane.  On Multipliers for Lagrange Problems , 1939 .

[150]  Manfred Morari,et al.  State-space interpretation of model predictive control , 1994, Autom..

[151]  Moritz Diehl,et al.  Handling uncertainty in economic nonlinear model predictive control: A comparative case study , 2014 .

[152]  Julio R. Banga,et al.  Improved scatter search for the global optimization of computationally expensive dynamic models , 2009, J. Glob. Optim..

[153]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[154]  Gerhard Reinelt,et al.  Direct methods with maximal lower bound for mixed-integer optimal control problems , 2009, Math. Program..

[155]  F. Clarke.  The Maximum Principle under Minimal Hypotheses , 1976 .

[156]  Mariano Giaquinta,et al.  Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105 , 1984 .

[157]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[158]  R. Sargent,et al.  Solution of a Class of Multistage Dynamic Optimization Problems. 2. Problems with Path Constraints , 1994 .

[159]  G. R. Sullivan,et al.  The development of an efficient optimal control package , 1978 .

[160]  R. Vinter,et al.  Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations , 1997 .

[161]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[162]  Stephen J. Wright,et al.  Optimization for Machine Learning , 2013 .

[163]  Jay H. Lee,et al.  Model predictive control: Review of the three decades of development , 2011 .

[164]  Olivier Bernard,et al.  Experiment selection for the discrimination of semi-quantitative models of dynamical systems , 2006, Artif. Intell..

[165]  Vassilios S. Vassiliadis,et al.  Heat exchanger network cleaning scheduling: From optimal control to mixed-Integer decision making , 2018, Comput. Chem. Eng..

[166]  Christodoulos A. Floudas,et al.  Deterministic Global Optimization in Nonlinear Optimal Control Problems , 2000, J. Glob. Optim..

[167]  W. Walter.  Differential and Integral Inequalities , 1970 .

[168]  S. Sager.  Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control , 2009 .