Application of a modified quasilinearization technique to totally singular optimal control problems

Optimal control has guided numerous applications in chemical engineering, and exact determination of optimal profiles is essential for operation of separation and reactive processes, and operating strategies and recipe generation for batch processes. Here, a simultaneous collocation formulation based on moving finite elements is developed for the solution of a class of optimal control problems. Novel features of the algorithm include the direct location of breakpoints for control profiles and a termination criterion based on a constant Hamiltonian profile. The algorithm is stabilized and performance is significantly improved by decomposing the overall nonlinear programming (NLP) formulation into an inner problem, which solves a fixed element simultaneous collocation problem, and an outer problem, which adjusts the finite elements based on several error criteria. This bilevel formulation is aided by a NLP solver (the interior point optimizer) for both problems as well as an NLP sensitivity component, which provides derivative information from the inner problem to the outer problem. This approach is demonstrated on 11 dynamic optimization problems drawn from the optimal control and chemical engineering literature.

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“…The modified singular optimal control problem is given by

\min_{u} z_{3} true(\frac{π}{2} true)

s . t . \frac{d z_{1}}{d t} = z_{2}; z_{1} (0) = 0

\frac{d z_{2}}{d t} = u; z_{2} (0) = 1

\frac{d z_{3}}{d t} = \frac{1}{2} z_{2}^{2} - \frac{1}{2} z_{1}^{2} + \hat{ε} u^{2}; z_{3} (0) = 0

- 1 \leq u \leq 1

with

\hat{ε} = 0.1

. The initial number of finite elements for Step 1 is 20 and the final number remains at 20.…”

Section: Optimal Control Of Modified Singular Problemsmentioning

confidence: 99%

A bilevel NLP sensitivity‐based decomposition for dynamic optimization with moving finite elements

Chen

Shao

Biegler³

2014

AIChE Journal

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“…(26) using the singular control given by Eq. (30) and store x,(tf)' (iv) Increment t l and go to step (i).…”

Section: Numerical Examplesmentioning

confidence: 99%

“…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: 99%

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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“…In this respect, we refer the reader to gradient techniques (Pagurek & Woodside, 1968), modified gradient techniques (Soliman & Ray, 1972), quasi Newton algorithm (Edge & Powers, 1976), quasi-linearization technique (Aly & Chan, 1973), indirect multiple shooting method (Aronna, Bonnans, & Martinon, 2013;Maurer, 1976), direct shooting method (Vossen, 2010), iterative dynamic programming method (Luus, 1992), line-up competition algorithm (Sun, 2010) and modified Pseudospectral method (Foroozandeh, Shamsi, Azhmyakov, & Shafiee, 2017). In most of the mentioned papers, the control structure is assumed a priori.…”

Section: Introductionmentioning

confidence: 99%

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

Foroozandeh

Shamsi

Pinho

2017

International Journal of Control

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This paper presents a new approach for the efficient and accurate solution of Singular Optimal Control Problems (SOCP). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of the solution. As the first step of this method, the SOCP is converted into a binary optimal control problem. Then, by utilizing the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the original optimal control and to compute the approximating solution of it (getting both the optimal state and control). The main advantages of the present method are that: (i) without a priori information, the structure of optimal control is detected; (ii) it produces good results even using a small number of collocation points; and (iii) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.

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Application of a modified quasilinearization technique to totally singular optimal control problems

Cited by 20 publications

References 7 publications

A bilevel NLP sensitivity‐based decomposition for dynamic optimization with moving finite elements

A bilevel NLP sensitivity‐based decomposition for dynamic optimization with moving finite elements

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

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

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