Pseudospectral Optimal Control and Its Convergence Theorems
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Summary. During the last decade, pseudospectral (PS) optimal control methods have emerged as demonstrable efficient methods for computational nonlinear optimal control. Some fundamental problems on the feasibility and convergence of the Legendre PS method are addressed. In the first part of this paper, we summarize the main results published separately in a series of papers on these topics. Then, a new result on the feasibility and convergence is proved. Different from existing results in the literature, in this new theorem neither the invertibility of necessary conditions nor the existence of limit points is assumed.
- K. Bollino, I. M. Ross, and D. Doman. Optimal nonlinear feedback guidance for reentry vehicles. Proc. of AIAA Guidance, Navigation and Control Conference, 2006.
- J.P. Boyd. Chebyshev and Fourier Spectral Methods. Dover, 2nd edition, 2001.
- C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Method in Fluid Dynamics. Springer-Verlag, New York, 1998.
- J. Cullum. Finite-dimensional approximations of state constrainted continuous optimal problems. SIAM J. Contr. Optimization, 10:649–670, 1972. CrossRef
- A.L. Dontchev and W.W. Hager. The Euler approximation in state constrained optimal control. Mathematics of Computation, 70:173–203, 2000. CrossRef
- G. Elnagar and M.A. Kazemi. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Computational Optimization and Applications, 11:195–217, 1998. CrossRef
- G. Elnagar, M.A. Kazemi, and M. Razzaghi. The pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. on Automat. Contr., 40:1793–1796, 1995. CrossRef
- F. Fahroo and I.M. Ross. Computational optimal control by spectral collocation with differential inclusion. Proc. of the 1999 Goddard Flight Mechanics Symposium, pages 185–200, 1999. NASA/CP-1999-209235.
- F. Fahroo and I.M. Ross. Costate estimation by a Legendre pseudospectral method. J. of Guidance, Control, and Dynamics, 24(2):270–277, 2001.
- F. Fahroo and I.M. Ross. Direct trajectory optimization by a Chebyshev pseudospectral method. J. of Guidance, Control, and Dynamics, 25(1):160–166, 2002.
- F. Fahroo and I.M. Ross. Pseudospectral methods for infinite-horizon nonlinear optimal control problems. Proc. of AIAA Guidance, Navigation and Control Conference, 2005.
- Q. Gong, W. Kang, and I.M. Ross. A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Trans. on Automat. Contr., 51(7):1115–1129, 2006. CrossRef
- W.W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik, 87:247–282, 2000. CrossRef
- S.I. Infeld and W. Murray. Optimization of stationkeeping for a libration point mission. AAS Spaceflight Mechanics Meeting, 2004. AAS 04-150.
- W. Kang, Q. Gong, and I.M. Ross. Convergence of pseudospectral methods for a class of discontinuous optimal control. Proc. of the 44rd IEEE Conf. on Decision and Contr., 2005.
- P. Lu, H. Sun, and B. Tsai. Closed-loop endoatmospheric ascent guidance. J. of Guidance, Control, and Dynamics, 26(2):283–294, 2003. CrossRef
- E. Polak. Optimization: Algorithms and Consistent Approximations. Springer-Verlag, Heidelberg, 1997.
- L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mischenko. The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York, 1962.
- J. Rea. Launch vehicle trajectory optimization using a Legendre pseudospectral method. Proc. of AIAA Guidance, Navigation and Control Conference, 2003. Paper No. AIAA 2003-5640.
- S.M. Robinson. Strongly regular generalized equations. Mathematics of Operations Research, 5:43–62, 1980.
- S.M. Robinson. An implicit function theorem for a class of nonsmooth functions. Mathematics of Operations Research, 16:292–309, 1991. CrossRef
- I.M. Ross and F. Fahroo. A unified framework for real-time optimal control. Proc. of the 42nd IEEE Conf. on Decision and Contr., 2003.
- I.M. Ross and F. Fahroo. Pseudospectral knotting methods for solving optimal control problems. J. of Guidance, Control, and Dynamics, 27(3):397–405, 2004.
- I.M. Ross and F. Fahroo. Issues in the real-time computation of optimal control. Mathematical and Computer Modelling, An International Journal, 43(9–10):1172–1188, 2006. CrossRef
- I.M. Ross, Q. Gong, F. Fahroo, and W. Kang. Practical stabilization through real-time optimal control. Proc. of the 2006 Amer. Contr. Conf., 2006.
- I.M. Ross, P. Sekhavat, A. Fleming, and Q. Gong. Pseudospectral feedback control: foundations, examples and experimental results. Proc. of AIAA Guidance, Navigation and Control Conference, 2006. AIAA-2006-6354.
- G. Sansone, A.H. Diamond, and E. Hille. Orthogonal Functions. Robert E. Krieger Publishing Co., Huntington, New York, 1977.
- A. Schwartz and E. Polak. Consistent approximations for optimal control problems based on Runge-Kutta integration. SIAM J. Contr. Optimization, 34:1235–1269, 1996. CrossRef
- P. Sekhavat, A. Fleming, and I.M. Ross. Time-optimal nonlinear feedback control for the NPSAT1 spacecraft. Proc. of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2005.
- S. Stanton, R. Proulx, and C. D’Souza. Optimal orbit transfer using a Legendre pseudospectral method. Proc. of AAS/AIAA Astrodynamics Specialist Conference, 2003. AAS-03-574.
- P. Williams, C. Blanksby, and P. Trivailo. Receding horizon control of tether system using quasi-linearization and Chebyshev pseudospectral approximations. Proc. of AAS/AIAA Astrodynamics Specialist Conference, 2003. AAS-03-534.
- Pseudospectral Optimal Control and Its Convergence Theorems
- Book Title
- Analysis and Design of Nonlinear Control Systems
- Book Subtitle
- In Honor of Alberto Isidori
- pp 109-124
- Print ISBN
- Online ISBN
- Springer Berlin Heidelberg
- Copyright Holder
- Additional Links
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- Editor Affiliations
- 1. Imperial College London
- 2. Universit`a di Roma Tor Vergata
- 3. University of Bologna
- Author Affiliations
- 4. Naval Postgraduate School, 93943, Monterey, CA, USA
- 5. Naval Postgraduate School, 93943, Monterey, CA, USA
- 6. Univerisity of Texas at San Antonio, 78249, San Antonio, TX, USA
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