A Hybrid Direct–Indirect Approach for Solving the Singular Optimal Control Problems of Finite and Infinite Order

Foroozandeh, Zahra; Shamsi, M.; Pinho, Maria do Rosário de

doi:10.1007/s40995-017-0176-2

Cited by 12 publications

(8 citation statements)

References 26 publications

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“…Multiple shooting has superior stability properties for highly nonlinear models, and also for problems where the solution of the IVP may be unbounded in the domain of interest compared with a single shooting strategy 63 . In chemical engineering, several studies have implemented shooting methods for the optimal and singular control, for example, reaction systems for biological and inorganic applications 64,65 and catalytic systems 66 …”

Section: Methodsmentioning

confidence: 99%

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

2022

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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.

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Section: Methodsmentioning

confidence: 99%

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

2022

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“…We set 10 −8 , 0.1, 0.7, 0.9 and 0.8 as the stopping tolerance, initial guesses for the switching times and co-states respectively for t f = 1. [19] 1.41 × 10 −08 17.90 --Hybrid Direct-Indirect Approach [12] 6.27 × 10 −09 -2.99 × 10 −09 1.14 × 10 −12 Control Parameterization [25] 1.41 × 10 −08 38.10 --Nested Simultaneous strategy [24] 1.41 × 10 −05 90 --ICRS method [5] 5.68 × 10 −05 40 --LARES mthod [16] 9.65 × 10 −04 ---Iterative Dynamic Programming [11] 5.68 × is not only efficient to reducing the required time for calculation, but also achieves the highest accuracy. Comparing different methods just based on CPU times may not sound reasonable as different platforms were used to calculate the solutions, however, adding absolute errors columns to show the accuracy of solutions based on their analytical ones can provide more efficient way to comparison.…”

Section: 1mentioning

confidence: 99%

The Switch Point Algorithm

Aghaee,

Hager

2020

Preprint

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The Switch Point Algorithm is a new approach for solving optimal control problems whose solutions are either singular or bang-bang or both, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. This class of problems can often be reduced to an optimization over the switching points. Formulas are derived for the derivative of the objective with respect to the switching points. Hence, gradient-based unconstrained optimization techniques including the conjugate gradient method or quasi-Newton methods can be used to solve the control problem. Three examples with known analytical solutions are used to illustrate the algorithm.

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“…The key idea is to use results of the direct method to initialize the indirect one. Recently Foroozandeh et al used this hybrid approach in the singular case. They begin obtaining a coarse solution with Euler transcription, then using this information they solve a boundary value problem (after applying Pontryagin's principle) with an adaptive indirect shooting method.…”

Section: Introductionmentioning

confidence: 99%

An efficient direct/indirect transcription approach for singular optimal control

et al. 2018

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It is very common in chemical engineering applications to find optimal control problems whose optimality conditions do not provide information about the control over an interval. This type of problems is called partially singular, as the control switches between nonsingular and singular arcs. When direct transcription is applied, the resulting nonlinear programming problem is ill conditioned. Some mesh refinement and rigorous iterative methods have been developed to determine the control profile and switching points. This work presents a practical alternative that quickly produces accurate state and control profiles without adding nonconvex terms. The problem is first solved with a large number of equally spaced finite elements. Then, unnecessary elements are removed while keeping the solution structure. Finally, direct and indirect approaches are combined to apply a regularization scheme only to the singular part. Seven examples were solved to test our strategy. Results provide good approximations to the analytical switching points.

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A Hybrid Direct–Indirect Approach for Solving the Singular Optimal Control Problems of Finite and Infinite Order

Cited by 12 publications

References 26 publications

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

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

The Switch Point Algorithm

An efficient direct/indirect transcription approach for singular optimal control

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