Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Abstract

In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

Selection and refinement of finite elements for optimal design and control: A Hamiltonian function approach

“…The optimization formulation in Equation ( 1) can be stated in some other equivalent forms, for example, Lagrange or Bolza formulations. 23 (1), the corresponding Hamiltonian function can be stated as follows:…”

“…In the objective function, the first term represents an economic term that only depends on static decision variables, for example, capital cost; whereas the second term accounts for transient costs in the process operation, for example, operating cost. The formulation in Equation (1) is stated as a “Mayer problem.” The optimization formulation in Equation (1) can be stated in some other equivalent forms, for example, Lagrange or Bolza formulations 23 . To transform Equation (1) into a Bolza formulation, differential Equation () is moved to the objective function as an integral term replacing function Ψ f ; whereas a Lagrange formulation requires only an integral term in the objective function, that is, no scalar terms such as function Φ( η ) is present in the objective function.…”

“…The implementation of other formulation is out of the scope of this study. Interested readers are encouraged to review 23 regarding the implementation of Pontryagin's minimum principle (PMP) in chemical engineering. The optimization Problem (1) can be discretized with OCFE and solved with conventional NLP solvers.…”

Integration of planning, scheduling, and control: A review and new perspectives

“…[71] Frameworks involving a nested decision-making structure comprising two loops have also been proposed for continuous and batch systems. [72][73][74][75] In the outer loop, rescheduling is carried out by solving an iSC problem (full-space or reduced order) to provide optimal setpoints values, which are tracked using either MPC or explicit MPC (eMPC) in the inner loop.…”

Bioprocess Intensification of a Continuous-Flow Enzymatic Bioreactor via Productivity Dynamic Optimization under Modeling Uncertainty

“…There are several error cost functions, like integrated time-weighted absolute error (ITAE), integrated squared error (ISE), integrated exponential time squared error (IETSE), integrated squared time cubed error (ISTCE), and integrated time exponential time squared error (ITETSE), that are employed for obtaining optimal controller parameters [19,20]. Also, iterative methods such as numerical optimization, fminsearch subroutine, and soft computing-based optimization methods, which include artificial bee colony (ABC) optimization, particle swarm optimization (PSO), genetic algorithm, cuckoo optimization, and quasi-opposition-based equilibrium optimizer, are used to minimize the cost functions [21][22][23][24][25].…”