Nonlinearly constrained time-delayed optimal control problems

Kaji, K.; Wong, K. H.

doi:10.1007/bf02191855

Cited by 19 publications

(15 citation statements)

References 13 publications

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“…Their theoretical justifications can be found in [21 ]. Also see [3] and [9] for further results on the control parameterization technique. For an illustration of the numerical results obtained by the control parameterization technique, see [21] and the references cited therein, where many practical problems are solved.…”

Section: Introductionmentioning

confidence: 97%

The control parameterization enhancing transform for constrained optimal control problems

Teo

Jennings

Lee

et al. 1999

J. Aust. Math. Soc. Series B, Appl. Math

139

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Consider a general class of constrained optimal control problems in canonical form. Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals. The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times. However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is. On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem. Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy. To overcome this difficulty, the switching points should also be taken as decision variables. This is the main motivation of the paper. A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms. For illustration, two non-trivial numerical examples are solved using the proposed method.

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Section: Introductionmentioning

confidence: 97%

The control parameterization enhancing transform for constrained optimal control problems

Teo

Jennings

Lee

et al. 1999

J. Aust. Math. Soc. Series B, Appl. Math

139

View full text Add to dashboard Cite

show abstract

“…We then use the unconstrained quasi-Newton method together with the simulated annealing technique to solve the dual problem of this approximated semi-infinite nonlinear programming problem. This method can usually overcome the numerical difficulties associated with the constraint transcription approach for the continuous state constraints used in [3], [10] and [11]. Also, our method requires fewer piece-wise constant controls to obtain a better numerical result than the constraint transcription method employed in [3], [10] and [11].…”

Section: Resultsmentioning

confidence: 94%

“…We note that the control parametrization technique in [3], [10] and [11] use constraint transcription to transform the infinite constraint into a standard constraint -an integration constraint. This method usually exhibits numerical difficulties for control problems.…”

Section: Numerical Examplementioning

confidence: 99%

Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints

Lee

Wong

2006

Optimization Methods and Software

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Consider the class of nonlinear time-delayed optimal control problems with continuous linear constraints. This class of problems is difficult to solve numerically. In this article, a computational method based on a semi-infinite programming approach is given. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the control, an approximated semi-infinite nonlinear problem is obtained. To solve this approximate problem, we use the dual parametrization method. The dual problem is a max-min problem, which can be solved by optimizing over two levels. The upper level can be solved by simulated annealing and the lower level can be solved by using any unconstrained optimisation technique, such as the quasi-Newton method.

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“…Although these papers develop algorithms to solve the problems, they do not provide convergence proof for the algorithms. References [12] and [10,13,15] presented formulations and algorithms for optimal control computations for linear and nonlinear time delay systems. Although these papers provide some error bound and convergence criteria, they do not develop expressions for the order of convergence.…”

Section: Introductionmentioning

confidence: 99%

O(h² ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations

Agrawal¹,

Gregory²

2003

SIAM J. Numer. Anal.

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The main purpose of this paper is to give numerical algorithms and the error analysis for delay quadratic problems in the calculus of variations. These methods are new, efficient, and accurate and have a global a priori error of O(h 2 ), where h is the distance between any two successive node points.We also derive the results for the general, numerical, delay problem, but we focus on proving our results in the simpler quadratic case since the extra technical numerical details have been given previously by the second author. In addition, the authors have previously shown how to reformulate general delay constrained problems in optimal control theory/constrained calculus of variations as unconstrained delay problems. Thus our numerical results and methods will hold for these general constrained problems also.Finally, we note that our algorithm, which solves the stationary condition(s) numerically, avoids the more difficult problems of piecing the solution of second order equations together and requires less smoothness in the solution. Thus we replace difficult second order boundary value problems with the easier task of approximating definite integrals involving first order derivatives.

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Nonlinearly constrained time-delayed optimal control problems

Cited by 19 publications

References 13 publications

The control parameterization enhancing transform for constrained optimal control problems

The control parameterization enhancing transform for constrained optimal control problems

Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints

O(h² ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations

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Nonlinearly constrained time-delayed optimal control problems

Cited by 19 publications

References 13 publications

The control parameterization enhancing transform for constrained optimal control problems

The control parameterization enhancing transform for constrained optimal control problems

Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints

O(h2 ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations

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O(h² ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations