Skip to main content
SpringerLink
Log in

A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems

  • Technical Note
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

A noniterative algebraic method is presented for solving differential Riccati equations which satisfy two-point boundary-value problems. This class of numerical problems arises in quadratic optimization problems where the cost functionals are composed of both continuous and discrete state penalties, leading to piecewise periodic feedback gains. The necessary condition defining the solution for the two-point boundary value problem is cast in the form of a discrete-time algebraic Riccati equation, by using a formal representation for the solution of the differential Riccati equation. A numerical example is presented which demonstrates the validity of the approach.

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. Geering, H. P.,Continuous-Time Optimal Control Theory for Cost Functionals Including Discrete State Penalty Terms, IEEE Transactions on Automatic Control, Vol. AC-21, No. 6, pp. 866–869, 1976.

    Google Scholar 

  2. Kwon, W. H., andPearson, A. E.,Linear Systems with Two-Point Boundary Lyapunov and Riccati Equations, IEEE Transactions on Automatic Control, Vol. AC-27, No. 2, pp. 436–441, 1982.

    Google Scholar 

  3. Kleinman, D. L.,On an Iterative Technique for Riccati Equation Computation, IEEE Transactions on Automatic Control, Vol. AC-13, No. 1, pp. 114–115, 1968.

    Google Scholar 

  4. Potter, J. E., andVan der Velde, W. E.,Optimum Mixing of Gyroscope and Star Tracker Data, Journal of Spacecraft and Rockets, Vol. 5, No. 5, pp. 536–540, 1968.

    Google Scholar 

  5. Turner, J. D., andChun, H. M.,Optimal Feedback Control of a Flexible Spacecraft during a Large-Angle Rotational Maneuver, Paper No. 82-1589-CP, AIAA Guidance and Control Conference, San Diego, California, 1982.

  6. Turner, J. D., Chun, H. M., andJuang, J.-N.,Closed-Form Solution for a Class of Optimal Quadratic Tracking Problems, Journal of Optimization Theory and Applications, Vol. 47, No. 4, pp. 465–481, 1985.

    Google Scholar 

  7. Potter, J. E.,Matrix Quadratic Solution, SIAM Journal on Applied Mathematics, Vol. 14, No. 3, pp. 496–501, 1964.

    Google Scholar 

  8. Jamshidi, M.,An Overview on the Solution of the Algebraic Matrix Riccati Equation and Related Problems, Large-Scale Systems: Theory and Application, Edited by M. G. Singh and A. P. Sage, North-Holland Publishing Company, Vol. 1, No. 3, pp. 167–192, 1980.

  9. Davison, E. J., The Numerical Solution of\(\dot X\) =A 1 X+XA 2+D, X(0) =C, IEEE Transactions on Automatic Control, Vol. AC-20, No. 4, pp. 566–567, 1975.

    Google Scholar 

  10. Serbin, S. M., andSerbin, C. A., A Time-Stepping Procedure for\(\dot X\) =A 1 X+XA 2+D, X(0) =C, IEEE Transactions on Automatic Control, Vol. AC-25, No. 6, pp. 1138–1141, 1980.

    Google Scholar 

  11. Pace, I. S., andBarnett, S.,Comparison of Numerical Methods for Solving Lyapunov Matrix Equations, International Journal of Control, Vol. 15, No. 5, pp. 907–915, 1972.

    Google Scholar 

  12. Bartels, R. D., andStewart, G. W.,Algorithm 432, A Solution of the Equation AX+XB=C, Communications of the Association for Computing Machinery, Vol. 15, No. 9, pp. 820–826, 1972.

    Google Scholar 

  13. Golub, G. H., Nash, S., andVan Loan, C.,A Hessenburg-Schur Method for the Problem AX+XB=C, IEEE Transactions on Automatic Control, Vol. AC-24, No. 6, pp. 909–913, 1979.

    Google Scholar 

  14. Vaughan, D. R.,A Nonrecursive Algebraic Solution for the Discrete Riccati Equation, IEEE Transactions on Automatic Control, Vol. AC-15, No. 5, pp. 597–599, 1970.

    Google Scholar 

  15. Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Halsted Press, Washington, DC, p. 47, 1975.

    Google Scholar 

  16. Sage, A. P.,Optimum Systems Control, Englewood Cliffs, New Jersey, p. 131, 1968.

    Google Scholar 

  17. O'Donnell, J. J.,Asymptotic Solution of the Matrix Riccati Equation of Optimal Control, Proceedings of the 4th Allerton Conference Circuit and System Theory, Urbana, Illinois, pp. 577–586, 1966.

Download references

Author information

Authors and Affiliations

  1. Control Theory/Dynamics Group, Cambridge Research Division of Photon Research Associates, Cambridge, Massachusetts

    H. M. Chun (Staff Scientist) & J. D. Turner (Leader)

Authors
  1. H. M. Chun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. D. Turner
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by W. E. Schmitendorf

The authors would like to thank Dr. Fernando Incertis, IBM Madrid Scientific Center, who reviewed this paper and pointed out that the two-point boundary-value necessary condition could be manipulated into the form of a discrete-time Riccati equation. His novel approach proved to be superior to the authors' previously proposed iterative continuation method.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chun, H.M., Turner, J.D. A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems. J Optim Theory Appl 51, 355–362 (1986). https://doi.org/10.1007/BF00939830

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00939830

Key Words

Search

Navigation