An indirect approach for singular optimal control problems

Andrés‐Martínez, Oswaldo; Biegler, Lorenz T.; Flores‐Tlacuahuac, Antonio

doi:10.1016/j.compchemeng.2020.106923

Cited by 18 publications

(14 citation statements)

References 31 publications

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“…Optimal control [23] strategies ensuring the domination of the strain of interest are derived using the Pontryagin maximum principle [24]. Since the system is affine w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Optimal Darwinian Selection of Microorganisms with Internal Storage

2022

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In this paper, we investigate the problem of species separation in minimal time. Droop model is considered to describe the evolution of two distinct populations of microorganisms that are in competition for the same resource in a photobioreactor. We focus on an optimal control problem (OCP) subject to a five-dimensional controlled system in which the control represents the dilution rate of the chemostat. The objective is to select the desired species in minimal-time and to synthesize an optimal feedback control. This is a very challenging issue, since we are are dealing with a ten-dimensional optimality system. We provide properties of optimal controls allowing the strain of interest to dominate the population. Our analysis is based on the Pontryagin Maximum Principle (PMP), along with a thorough study of singular arcs that is crucial in the synthesis of optimal controls. These theoretical results are also extensively illustrated and validated using a direct method in optimal control (via the Bocop software for numerically solving optimal control problems). The approach is illustrated with numerical examples with microalgae, reflecting the complexity of the optimal control structure and the richness of the dynamical behavior.

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“…Optimal control [23] strategies ensuring the domination of the strain of interest are derived using the Pontryagin maximum principle [24]. Since the system is affine w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Optimal Darwinian Selection of Microorganisms with Internal Storage

2022

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“…The drawback of the UTM is that it utilizes regularization for singular controls, thereby generating results that are not as accurate as the other direct methods or the indirect approach devised in Ref. 26. For brevity, the direct/indirect transcription strategy and the indirect approach proposed in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Andrés‐Martínez et al then proposed an indirect approach in Ref. 26, which is initialized using a direct method. The optimality conditions lead to a two‐point boundary‐value problem (TPBVP) along with complementary constraints, similar to the traditional indirect methods.…”

Section: Introductionmentioning

confidence: 99%

“…The UTM is simpler than the indirect approach proposed in Ref. 26 since no additional complementarity conditions are needed. Furthermore, as compared to this indirect approach, an analytical control law is obtained while using the UTM, which provides more insight about the solution structure.…”

Section: Introductionmentioning

confidence: 99%

“…For brevity, the direct/indirect transcription strategy and the indirect approach proposed in Refs. 22 and 26, respectively, have not been included in this study for results and comparison with the UTM. Both these methods, however, offer unique features that are highly desirable for solving challenging OCPs with singular control arcs.…”

Section: Introductionmentioning

confidence: 99%

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Solving singular control problems using uniform trigonometrization method

2021

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This study investigates the application of a recently developed construct, the uniform trigonometrization method (UTM), to the singular control problems in chemical engineering. The UTM involves minimal modifications to the original problem, thereby generating near-singular control solutions that can be used for conceptual design and serve as an alternate to direct techniques like nested and simultaneous approaches.Eight classical singular control problems with known analytical solutions and three complex singular control problems from chemical engineering domain are solved in this study. The results obtained using the UTM for these problems are found to match well with the literature and are of higher resolution as compared to the results obtained using a direct pseudospectral-based solver. The ability of the UTM to handle complex chemical engineering problems with both singular controls and state path constraints has also been demonstrated in this study.

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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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An indirect approach for singular optimal control problems

Cited by 18 publications

References 31 publications

Optimal Darwinian Selection of Microorganisms with Internal Storage

Optimal Darwinian Selection of Microorganisms with Internal Storage

Solving singular control problems using uniform trigonometrization method

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

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