Skip to main content
SpringerLink
Log in

Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

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

Abstract

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.

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. S. Abarbanel and D. Gottlieb, “Information content in spectral calculation,” Progress in Scientific Computing, vol. 6, pp. 345-355, 1984.

    Google Scholar 

  2. R. Asky, “Mean convergence of orthogonal series and Lagrange interpolation,” Acta Math. Acad. Sci. Hungar, vol. 23, pp. 71-85, 1972.

    Google Scholar 

  3. J.T. Betts and W.P. Huffman, “Path-constrained trajectory optimization using sparse sequential quadratic programming,” J. Guidance, Control, and Dynamics, vol. 16, no.1, pp. 59-68, 1989.

    Google Scholar 

  4. R.R. Bless, D.H. Hodges, and H. Seywald, “Finite element method for the solution of state-constrained optimal control problems,” J. Guidance, Control, and Dynamics, vol. 18, no.5, pp. 1036-1043, 1995.

    Google Scholar 

  5. A.E. Bryson and Y.C. Ho, Applied Optimal Control, Hemisphere: New York, 1975.

    Google Scholar 

  6. R. Bulirsch and D. Kraft (Eds.), Computational Optimal Control, Birkhauser: Boston, 1994.

    Google Scholar 

  7. W. Cai, D. Gottlieb, and A. Harten, “Cell-averaging Chebyshev methods for hyperbolic problems,” Comput. Math. Appl., vol. 24, nos.5/6, pp. 34-49, 1992.

    Google Scholar 

  8. C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Varlag: New York, 1988.

    Google Scholar 

  9. J.E. Cuthrell and L.T. Biegler, “On the optimization of differential-algebraic process systems,” AICHE Journal, vol. 3, nos.1/2, pp. 1257-1270, 1987.

    Article  Google Scholar 

  10. J.E. Cuthrell and L.T. Biegler, “Simultaneous optimization and solution methods for batch reactor control profiles,” Computers Chem. Eng., vol. 13, pp. 49-62, 1989.

    Article  Google Scholar 

  11. P.J. Davis and Rabinowitz, Methods of Numerical Integration, Academic Press: London, 1984.

    Google Scholar 

  12. G.N. Elnagar and M.A. Kazemi, “A cell-averaging Chebyshev spectral method for the controlled duffing oscillator,” Appl. Numer. Math., vol. 18, pp. 461-471, 1995.

    Article  Google Scholar 

  13. G.N. Elnagar and M.A. Kazemi, “A cell-averaging Chebyshev spectral method for nonlinear Fredholm-Hammerstein integral equations,” Intern. J. Computer Mathematics, vol. 60, pp. 91-104, 1996.

    Google Scholar 

  14. G.N. Elnagar, M.A. Kazemi, and M. Razzaghi, “The pseudospectral legendre method for discretizing optimal control problems,” IEEE Trans. Automat. Cont., vol. 40, no.10, pp. 1793-1796, 1995.

    Article  Google Scholar 

  15. P.E. Gill, W. Murray, M.A. Saunders, and M.H. Wright, User's Guide for NZOPT 1.0: A Fortran Package for Nonlinear Programming, McDonnell Douglas Aerospace, Huntington Beach, CA, April 1993.

    Google Scholar 

  16. W.W. Hager, “Multiplier methods for nonlinear optimal control problems,” SIAM J. Numerical Anal., vol. 37, pp. 1061-1080, 1990.

    Google Scholar 

  17. A.L. Herman and B.A. Conway, “Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules,” J. Guidance, Control, and Dynamics, vol. 19, no.3, pp. 592-599, 1996.

    Google Scholar 

  18. M.L. Nagurka, “Fourier-based optimal control of nonlinear dynamic systems,” J. Dyn. Syst., Measur., and Control, vol. 112, pp. 17-26, 1990.

    Google Scholar 

  19. E. Polak et al., “A Method of Centers Based on Barrier Function Methods for Solving Optimal Control Problems with Continum State and Control Constraints,” SIAM J. Optim., vol. 31, pp. 159-179, 1993.

    Google Scholar 

  20. K. Schittkowskki, “NLPQL: A Fortran subroutine for solving constrained nonlinear programming problems,” Operations Research Annals, vol. 5, pp. 385-400, 1985.

    Google Scholar 

  21. H. Seywald and R.R. Kumar, “Finite difference scheme for automatic costate calculation,” J. Guidance, Control, and Dynamics, vol. 19, no.1, pp. 231-239, 1996.

    Google Scholar 

  22. K.L. Teo and K.H. Wong, “Nonlinearly constrained optimal control problems,” J. Austral. Math Soc., Ser. B, vol. 33, pp. 507-530, 1992.

    Google Scholar 

  23. J. Vlassenbroeck and R.V. Dooren, “A Chebyshev technique for solving nonlinear optimal control problems,” IEEE Tran. Automat. Cont., vol. 33, no.4, pp. 333-340, 1988.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of South Carolina Spartanburg, Spartanburg, SC, 29303

    Gamal N. Elnagar

  2. Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina, 28223

    Mohammad A. Kazemi

Authors
  1. Gamal N. Elnagar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad A. Kazemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Elnagar, G.N., Kazemi, M.A. Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems. Computational Optimization and Applications 11, 195–217 (1998). https://doi.org/10.1023/A:1018694111831

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018694111831

Search

Navigation