Skip to main content
SpringerLink
Log in

Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albassam, B.A.: Optimal near-minimum-time control design for flexible structures. J. Guid. Control Dyn. 25(4), 618–625 (2002)

    Article  Google Scholar 

  2. Betts, J.T., Erb, S.O.: Optimal low thrust trajectories to the Moon. SIAM J. Appl. Dyn. Syst. 2(2), 144–170 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  4. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001)

    MATH  Google Scholar 

  5. Bryson, A.E.: Dynamic Optimization. Addison-Wesley Longman, Reading (1999)

    Google Scholar 

  6. Bryson, A.E., Ho, Y.C.: Applied Optimal Control. Hemisphere, New York (1975)

    Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Method in Fluid Dynamics. Springer, New York (1988)

    Google Scholar 

  8. Chryssoverghi, I., Coletsos, J., Kokkinis, B.: Discretization methods for optimal control problems with state constraints. J. Comput. Appl. Math. 191, 1–31 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dontchev, A.L.: Discrete approximations in optimal control. In: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. IMA Vol. Math. Appl., vol. 78, pp. 59–81. Springer, New York (1996). MR 97h:49043

    Google Scholar 

  10. Dontchev, A.L., Hager, W.W.: A new approach to Lipschitz continuity in state constrained optimal control. Syst. Control Lett. 35(3), 137–143 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control. Math. Comput. 70, 173–203 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal. 38(1), 202–226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Elnagar, G., Kazemi, M.A., Razzaghi, M.: The pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40, 1793–1796 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Elnagar, G., Kazemi, M.A.: Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Comput. Optim. Appl. 11, 195–217 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Enright, P.G., Conway, B.A.: Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. J. Guid. Control Dyn. 15(4), 994–1002 (1992)

    Article  MATH  Google Scholar 

  16. Fahroo, F., Ross, I.M.: Costate estimation by a Legendre pseudospectral method. AIAA J. Guid. Control Dyn. 24(2), 270–277 (2001)

    Article  Google Scholar 

  17. Fleming, A.: Real-time optimal slew maneuver design and control. Astronautical Engineer’s Thesis, US Naval Postgraduate School, December 2004

  18. Freud, G.: Orthogonal polynomials. Pergamon Press, Elmsford (1971)

    Google Scholar 

  19. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4), 979–1006 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gong, Q., Kang, W., Ross, I.M.: A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Trans. Autom. Control 51(7), 1115–1129 (2006)

    Article  MathSciNet  Google Scholar 

  21. Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hager, W.W.: Numerical analysis in optimal control. In: Hoffmann, K.-H., Lasiecka, I., Leugering, G., Sprekels, J., Troeltzsch, F. (eds.) International Series of Numererical Mathematics, vol. 139, pp. 83–93. Birkhäuser, Basel (2001)

    Google Scholar 

  23. Hargraves, C.R., Paris, S.W.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dyn. 10, 338–342 (1987)

    Article  MATH  Google Scholar 

  24. Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hawkins, A.M., Fill, T.R., Proulx, R.J., Feron, E.M.: Constrained trajectory optimization for Lunar landing. In: AAS Spaceflight Mechanics Meeting, AAS 06-153, Tampa, FL, January 2006

  26. Infeld, S.I., Murray, W.: Optimization of stationkeeping for a Libration point mission. In: AAS Spaceflight Mechanics Meeting, AAS 04-150, Maui, HI, February 2004

  27. Josselyn, S., Ross, I.M.: A rapid verification method for the trajectory optimization of Reentry vehicles. J. Guid. Control Dyn. 26(3), 505–508 (2003)

    Article  Google Scholar 

  28. Kang, W., Gong, Q., Ross, I.M.: Convergence of pseudospectral methods for nonlinear optimal control problems with discontinuous controller. In: 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05), Seville, Spain, pp. 2799–2804 (2005)

  29. Lu, P., Sun, H., Tsai, B.: Closed-loop endoatmospheric ascent guidance. J. Guid. Control Dyn. 26(2), 283–294 (2003)

    Article  Google Scholar 

  30. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) Series, vol. 330. Springer, Berlin (2005)

    MATH  Google Scholar 

  31. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II: Applications. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) Series, vol. 331. Springer, Berlin (2005)

    MATH  Google Scholar 

  32. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    MATH  Google Scholar 

  33. Paris, S.W., Hargraves, C.R.: OTIS 3.0 Manual. Boeing Space and Defense Group, Seattle (1996)

    Google Scholar 

  34. Pesch, H.J.: A practical guide to the solution of real-life optimal control problems. Control Cybern. 23, 7–60 (1994)

    MATH  MathSciNet  Google Scholar 

  35. Polak, E.: A historical survey of computational methods in optimal control. SIAM Rev. 15, 553–548 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  36. Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  37. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    MATH  Google Scholar 

  38. Rea, J.: Launch vehicle trajectory optimization using a Legendre pseudospectral method. In: Proceedings of the AIAA Guidance, Navigation and Control Conference, Paper No. AIAA 2003-5640, Austin, TX, August 2003

  39. Riehl, J.P., Paris, S.W., Sjauw, W.K.: Comparision of implicit integration methods for solving aerospace trajectory optimization problems. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA Paper No. 2006-6033, Keystone, CO, 21–24 August 2006

  40. Ross, I.M.: A historical introduction to the covector mapping principle. In: AAS/AIAA Astrodynamics Specialist Conference, Paper AAS 05-332, Tahoe, NV, 8–11 August 2005

  41. Ross, I.M.: A roadmap for optimal control: The right way to commute. In: Annals of the New York Academy of Sciences, vol. 1065. New York, NY, January 2006

  42. Ross, I.M.: User’s manual for DIDO: A MATLAB application package for solving optimal control problems. Elissar LLC., Technical Report TR-705 (2007)

  43. Ross, I.M., Fahroo, F.: A pseudospectral transformation of the covectors of optimal control systems. In: Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001

  44. Ross, I.M., D’Souza, C.N.: Hybrid optimal control framework for mission planning. J. Guid. Control Dyn. 28(4), 686–697 (2005)

    Article  Google Scholar 

  45. Ross, I.M., Fahroo, F.: Discrete verification of necessary conditions for switched nonlinear optimal control systems. In: Proceedings of the American Control Conference, Boston, MA, June 2004

  46. Ross, I.M., Fahroo, F.: Pseudospectral knotting methods for solving optimal control problems. AIAA J. Guid. Control Dyn. 27(3), 397–405 (2004)

    Article  Google Scholar 

  47. Ross, I.M., Fahroo, F.: Pseudospectral methods for optimal motion planning of differentially flat systems. IEEE Trans. Autom. Control 49(8), 1410–1413 (2004)

    Article  MathSciNet  Google Scholar 

  48. Ross, I.M., Fahroo, F.: Legendre pseudospectral approximations of optimal control problems. In: Kang, W. et al. (eds.) Lecture Notes in Control and Information Sciences, vol. 295, pp. 327–342. Springer, New York (2003)

    Google Scholar 

  49. Stanton, S., Proulx, R., D’Souza, C.N.: Optimal orbit transfer using a Legendre pseudospectral method. In: AAS/AIAA Astrodynamics Specialist Conference, AAS-03-574, Big Sky, MT, 3–7 August 2003

  50. Strizzi, J., Ross, I.M., Fahroo, F.: Towards real-time computation of optimal controls for nonlinear systems. In: Proc. AIAA GNC Conf., Monterey, CA, August 2002

  51. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  52. Veliov, V.M.: On the time-discretization of control systems. SIAM J. Control Optim. 35(5), 1470–1486 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  53. Vinter, R.: Optimal Control,. Birkhäuser, Boston (2000)

    Google Scholar 

  54. von Stryk, O.: Numerical solution of optimal control problems by direct collocation. In: Bulirsch, R. et al. (eds.) Optimal Control: Calculus of Variations, Optimal Control Theory and Numerical Methods. Birkhäuser, Boston (1993)

    Google Scholar 

  55. Williams, P., Blanksby, C., Trivailo, P.: Receding horizon control of tether system using quasilinearization and Chebyshev pseudospectral approximations. In: AAS/AIAA Astrodynamics Specialist Conference, Paper AAS 03-535, Big Sky, MT, 3–7 August 2003

  56. Yan, H., Alfriend, K.T.: Three-axis magnetic attitude control using pseudospectral control law in eccentric orbits. In: AAS Spaceflight Mechanics Meeting, AAS 06-103, Tampa, FL, January 2006

Download references

Author information

Authors and Affiliations

  1. Department of Electrical & Computer Engineering, University of Texas at San Antonio, San Antonio, TX, 78249-1644, USA

    Qi Gong

  2. Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA, 93943, USA

    I. Michael Ross

  3. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA

    Wei Kang & Fariba Fahroo

Authors
  1. Qi Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Michael Ross
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fariba Fahroo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Gong.

Additional information

The research was supported in part by NPS, the Secretary of the Air Force, and AFOSR under grant number, F1ATA0-60-6-2G002.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gong, Q., Ross, I.M., Kang, W. et al. Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control. Comput Optim Appl 41, 307–335 (2008). https://doi.org/10.1007/s10589-007-9102-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9102-4

Keywords

Search

Navigation