[논문]A Comparative Study of Transcription Techniques for Nonlinear Optimal Control Problems Using a Pseudo-Spectral Method

Kim, Chang-Joo; Sung, Sangkyung

doi:10.5139/ijass.2015.16.2.264

A Comparative Study of Transcription Techniques for Nonlinear Optimal Control Problems Using a Pseudo-Spectral Method 원문보기

International journal of aeronautical and space sciences, v.16 no.2, 2015년, pp.264 - 277

Kim, Chang-Joo (Department of Aerospace and Information Engineering, Konkuk University) , Sung, Sangkyung (Department of Aerospace and Information Engineering, Konkuk University)

Abstract ▼ AI-Helper

This article investigates various transcription techniques for the Legendre pseudospectral (PS) method to compare the pros and cons of each approach. Eight combinations from four different types of collocation points and two discretization methods for dynamic constraints, which differentiate Legendre PS transcription techniques, are implemented to solve a carefully selected test set of nonlinear optimal control problems (NOCPs). The convergence property and prediction accuracy are compared to provide a useful guideline for selecting the best combination. The tested NOCPs consist of the minimum time, minimum energy, and problems with state and control constraints. Therefore, the results drawn from this comparative study apply to the solution of similar types of NOCPs and can mitigate much debate about the best combinations. Additionally, important findings from this study can be used to improve the numerical efficiency of the Legendre PS methods. Three PS applications to the aerospace engineering problems are demonstrated to prove this point.

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문제 정의

The results of the example applications of the PS method to the aerospace engineering problems show that the PS methods are efficiently applicable in analyzing the complex and large-scale aerospace problems. In addition, the selection of the best combination among the collocation points and the trascription methods can be cruicial as investigated through this paper.
This paper focuses on investigating the pros and cons of various transciption techniques for the Legendre PS method. Comparative studies were conducted using six carefully seclected NOCP test problems to identify the best combination of collocation point types and of time integration methods.

가설 설정

i) The norm of the Lagrangian function gradient for the NLP is less than 10-4.

제안 방법

In parallel with these studies, various brand-new algorithms were proposed to efficiently handle the large scale problems, to enhance computational efficiency, and to remove the limitations posed by the standard PS approaches [18-20]. All these studies utilized the important findings of the present study and the integration methods are applied using the LGL collocation points. Consequently, the main results of the studies are recapitulated to provide useful guideline for the efficient application of PS to the aerospace problems.
In a case when the accurate estimation of the states and controls around the initial points is required as in the model predictive control, the PS method with slow convegence in predicting the intial controls can provide unstable divergent control inputs to the system. Although the study of the costate estimations will be a topic of a future study, the results of the present study of various types of test problems can be used for the best selection of transcription techniques with the PS method. Finally, the applications of the PS methods to the trajectory optimization for the aircaft and spacecraft were demonstrated and the results denote that the PS method can be a useful tool in solving complex aerospace engineering problems.
This paper investigates the effect of each selectable combination of transcription techniques on numerical convergence and solution accuracy of the Legendre PS method. Because the numerical characteristics of each approach are highly affected by the type of NOCP and by the constraints, we carefully selected six test problems with analytic solutions from various studies in the literature [7,10-12] to encompass a wide range of NOCPs, such as the minimum time, minimum energy, and problems with or without state and control constraints. Convergence history and accuracy in predicting the states and controls for each problem are throughly investigated using the results with applicable combinations.
Convergence history and accuracy in predicting the states and controls for each problem are throughly investigated using the results with applicable combinations. Therefore, the results of this study can be used to identify the numerical advantages and disadvantages of each combination among transcription techniques, from which the best one can be selected considering each characteristics of the NOCP. Finally, various applications of the PS methods to the aircraft and spacecraft flight dynamics are demonstrated to show that the PS method can be a useful tool in solving complex aerospace engineering problems and in designing an optimal trajectory.
This paper investigates the effect of each selectable combination of transcription techniques on numerical convergence and solution accuracy of the Legendre PS method. Because the numerical characteristics of each approach are highly affected by the type of NOCP and by the constraints, we carefully selected six test problems with analytic solutions from various studies in the literature [7,10-12] to encompass a wide range of NOCPs, such as the minimum time, minimum energy, and problems with or without state and control constraints.
This study intends to use a high-fidelity rotorcraft model, which typically includes fast rotor dynamics, in the NOCP analyses to accurately predict the emergency procedures of the rotorcraft after the engine failure.

이론/모형

The NLP formulated both for the differentiation and integration methods are solved using the robust SQP(rSQP) algorithm proposed by Fabien [13]. The algorithmic details are illustrated in [14] with its applications to both the example NLP test problems and to the NOCP analyses.
In this strategy, Legendre or Chebyshev interpolating polynomials are typically used to approximate states and controls in the continuous time domain. This paper focuses on a method based on Legendre polynomials for PS collocation. The set of collocation points, which are typically

성능/효과

(iii) With any of LG, LGR, FLGR, and LGL points, the PS method presents extremely slow convergence around two end points.

후속연구

All these studies utilized the important findings of the present study and the integration methods are applied using the LGL collocation points. Consequently, the main results of the studies are recapitulated to provide useful guideline for the efficient application of PS to the aerospace problems.

참고문헌 (23)

Kim, C.-J., Sung, S., Park, S. H. and Jung, S.-N., "Selection of Rotorcraft Models for Application to Optimal Control Problems," Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, 2008, pp. 1386-1399.

상세보기
Kim, C.-J., Lee, J., Byun, Y. H. and Yu, Y. H., "Nonlinear optimal control analysis of helicopter maneuver problems using the indirect method," Transactions of the Japan Society for Aeronautical and Space Sciences, Vol. 51, No. 171, 2008, pp. 43-51.

상세보기
Kim, C.-J., Sung, S., Yang, C. D. and Yu, Y. H., "Rotorcraft Trajectory Tracking Using the Sate-Dependent Riccati Equation Controller," Transactions of the Japan Society for Aeronautical and Space Sciences, Vol. 51, No. 173, 2008, pp. 184-192.

상세보기
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상세보기
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상세보기
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상세보기
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상세보기
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상세보기
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