- Volume 16 Issue 2
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.
Collocation Points;Gauss Quadrature;Nonlinear Optimal Control;Pseudospectral Transcription Techniques
- 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. https://doi.org/10.2322/tjsass.51.43
- 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. https://doi.org/10.2322/tjsass.51.184
- Kim, C.-J., Lee, J. H., Park, S. H., Sung, S. and Jung, S.-N., "Nonlinear Optimal Control Analysis Using State-Dependent Matrix Exponential and Its Integrals," Journal of Guidance, Control, and Dynamics, Vol. 31, No. 5, pp. 1386-1399, 2008. https://doi.org/10.2514/1.33212
- Benson, D., A Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, MIT, Department of Aeronautics and Astronautics, November 2004.
- Huntington, G. T., Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control Problems, Ph.D. Thesis, MIT, Department of Aeronautics and Astronautics, June 2007
- Darby, C. L., HP-Pseudo-spectral Method for Solving Continuous-Time Nonlinear Optimal Control Problems, Ph. D. Thesis, University of Florida, May 2011.
- D. Garg, M. Patterson, W. W. Hager, A. V. Rao, D. A. Benson, and G. T. Huntington, "A Unified Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral methods," Automatica, Published Online, August 2010. DOI:10.1016/j.automatica.2010.06.048. https://doi.org/10.1016/j.automatica.2010.06.048
- Williams, P., "A Gauss-Lobatto Quadrature Method for Solving Optimal Control Problems," Australian and New Zealand Industrial and Applied Mathematics Journal, Vol. 47, 2005, pp. 101-115.
- Bryson, A. E., Jr. and Ho, Y. C., Applied Optimal Control, Hemisphere Publishing, 1975.
- Kirk, D. E., Optimal Control Theory; An Introduction, Dover, New York, 1970.
- Fabien, B. C., "Implementation of a robust SQP algorithm," Optimization Method & Software, Vol. 23, No. 6, October 2008, pp. 827-846. https://doi.org/10.1080/10556780801996228
- Kim, C.-J., Sung, S. and Shin, K., "Pseudo-spectral Application to Nonlinear Optimal Trajectory Generation of a Rotorcraft," The First International Conference on Engineering and Technology Innovation, Kenting, Taiwan, November 11-15, 2011.
- 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. https://doi.org/10.2514/1.33212
- Kim, C.-J. et al., "Flight Dynamics Analyses of a Propeller-Driven Airplane (I): Aerodynamic and Inertial Modeling of the Propeller," IJASS, Vol. 15, No. 4, pp. 345-355, 2014.
- Kim, C.-J. et al., "Flight Dynamics Analyses of a Propeller-Driven Airplane (II): Building a High-Fidelity Mathematical Model and Applications," IJASS, Vol. 15, No. 4, pp. 356-365, 2014.
- Lee, D. H., Hur, S. U. and Kim, C.-J., "Study on Optimum Lunar Landing Trajectories Considering Initial Altitude of Lunar Parking Orbit," KSASS Autumn Conference, Jeju, Korea, November, 2014.
- Kim, C.-J., Sung, S., Park, S. H., Jung, S. N. and Park, T. S., "Numerical Time Scale Separation for Nonlinear Optimal Control Analyses with Applications to Rotorcrafts," AIAA Journal of Guidance, Control and Dynamics, Vol. 37, No. 2, March-April 2014, pp. 658-673. https://doi.org/10.2514/1.59557
- Kim, C.-J., Park, S. H., S.-N. Jung, and Sung, S., "Generalized Computational Nodes for Pseudospectral Methods," IJASS, Vol. 15, No. 2, 2014, pp. 183-189.
- Kim, C.-J. and Sung, S., "Efficient ST Techniques for Nonlinear Optimal Control Analyses Using a Pseudospectral Framework," IEEE Transactions on Control Systems Technology, Vol. 23, No. 3, May 2015, pp. 1110-1116. https://doi.org/10.1109/TCST.2014.2346994
- Park, T. S., Research on Dynamic Modeling, Flight Dynamic Analysis and Optimum Trajectory Generation, M.S. Thesis, Department of Aerospace Information Systems Engineering, Konkuk University, Seoul, Korea, December, 2013.
- Yun, Y.-H., Kim, C.-J., Shin, K.-C., Yang, C.-D. and Cho, I.-J., "Building the Flight Dynamic Analysis Program, HETLAS, for the Development of Helicopter FBW System," 1st Asian Australian Rotorcraft Forum and Exhibition 2012, Bexco, Busan, Korea, February 12-15, 2012.
- Kim, C.-J., Shin, K.-C., Yang, C.-D. and Cho, I.-J., "Interface Features of Flight Dynamic Analysis Program, HETLAS, for the Development of Helicopter FBW System," 1st Asian Australian Rotorcraft Forum and Exhibition 2012, Bexco, Busan, Korea, February 12-15, 2012.
- Singh, B., A Weighted Residual Framework for Formulation and Analysis of Direct Transcription Methods for Optimal Control, Ph.D. Thesis, Texas A&M University, December 2010.
연구 과제 주관 기관 : Konkuk University