The computation of optimal singular control

Aly, Gamal M.

doi:10.1080/00207177808922489

Cited by 18 publications

(17 citation statements)

References 11 publications

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“…This is a linear-quadratic regulator problem described by Aly. 10 The integral of the sum of the squares of the position and speed of a mobile unit over a fixed time interval is minimized min 1 2…”

Section: Aly Problemmentioning

confidence: 99%

“…9 Aly extended the method of Anderson 9 by using the modified quasilinearization method to solve this class of singular optimal control problems. 10 Maurer also assumed that the solution structure and explicit expression for singular control could be determined beforehand. The resulting enlarged boundary value problem for singular control problems was solved by a multiple shooting method.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the formulation coincides with the approach of Maurer 11 for the bang-singular-bang case. 12 All of these approaches [9][10][11][12] solve the boundary value problem given by the necessary conditions, assuming a known solution structure and explicit expression for the singular control. In practice, however, it is difficult to determine the correct switching structure of the solution beforehand.…”

Section: Introductionmentioning

confidence: 99%

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Nested direct transcription optimization for singular optimal control problems

Chen

Biegler

2016

AIChE Journal

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in Wiley Online Library (wileyonlinelibrary.com)Singular optimal control problems arise frequently on a broad range of chemical engineering applications. Determination of the accurate dynamic structure of the optimal solution profile and the junctions between optimal nonsingular and singular arcs is essential for operating strategies, as well as equipment designs for many processes. In a previous study (Chen, Shao, and Biegler, AIChE J. 60(3), pp. 966-979, 2014) a nested optimization formulation that finds the optimal mesh distribution and determines exact control profiles for nonsingular optimal control problems without state constraints is developed. This study extends this approach to singular optimal control problems. To satisfy the necessary optimality conditions for singular optimal control problems with a well-defined solution strategy, the overall nonlinear programming formulation resulting from direct transcription is decomposed into inner and outer problems. The key feature of this algorithm is that it converges to a solution that satisfies the discretized Euler-Lagrange equations of the original singular optimal control problem; we prove this under suitable assumptions. This is obtained through the introduction of pseudomultipliers that reconstruct the necessary optimality conditions for singular optimal control in the outer problem. We demonstrate this approach on eight classical singular control problems with known solutions, as well as three larger singular control problems derived from chemical engineering applications. V C 2016 American Institute of Chemical Engineers AIChE J, 62: [3611][3612][3613][3614][3615][3616][3617][3618][3619][3620][3621][3622][3623][3624][3625][3626][3627] 2016 Here, rðs; v i Þ are monotonic functions that may take polynomial, rational or exponential forms and v i is a vector of decision variables of dimension less than K.For the Aly-Chan problem, substituting constraint (16) into (14), with u(t) represented as a constant in each element leads

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Section: Aly Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nested direct transcription optimization for singular optimal control problems

Chen

Biegler

2016

AIChE Journal

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“…These conditions can be utilized to obtain an analytic solution for simple problems described by a few dynamic equations (1-3). However, for problems described by a large number of dynamic equations [3,4] Stutts (23) Thomas (18] Jacobson [24] Edgar and Lapidus [25,26] Maurer [27] Oberele (28) Aly (29) Aly and Chan [30] Aly and Megeed (31) Soliman and Ray (32) Kumar [33) Jacobson [34] Cuthrell and Biegler [29] Downloaded by [University of California Santa Barbara] at 04: 20 17 June 2016 SINGULAR CONTROL PROBLEMS 167 (say 4 or more) numerical solutions are inevitable. There have been a number of numerical techniques proposed for solving singular control problems, which are summarized in Table II.…”

Section: Introductionmentioning

confidence: 98%

An Improved Computational Algorithm for Singular Control Problems in Chemical Reaction Engineering

Modak

Bonte

Lim

1989

Chemical Engineering Communications

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A unidirectional method is developed to solve optimal singular control problems with bounded state variables. The computational algorithm utilizes the necessary conditions of optimality and an unidirectional scheme to reduce the iterations required in two-point boundary value problems. The advantage of the method is that it satisfies all the necessary conditions of optimality and results in a considerable reduction in computational effort. Three numerical examples illustrate the use of these algorithms for solving several chemical reaction engineering problems.

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“…A common strategy, known as the indirect approach, first applies Pontryagin's minimum principle and then solves numerically the resulting differential algebraic equation (DAE) system with an explicit expression for the control variable obtained by reducing the index of the DAE . Several variants of this approach have been developed using different shooting methods and modified quasilinearization . The disadvantages of this strategy are that knowledge of the switching structure is required beforehand, which is difficult to derive previously and the expression for the control can be hard to obtain even for small problems.…”

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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The computation of optimal singular control

Cited by 18 publications

References 11 publications

Nested direct transcription optimization for singular optimal control problems

Nested direct transcription optimization for singular optimal control problems

An Improved Computational Algorithm for Singular Control Problems in Chemical Reaction Engineering

An efficient direct/indirect transcription approach for singular optimal control

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