Advertisement

Journal of Optimization Theory and Applications

, Volume 51, Issue 2, pp 355–362 | Cite as

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

  • H. M. Chun
  • J. D. Turner
Technical Note

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.

Key Words

Optimal control piecewise periodic feedback gains two-point boundary-value problems continuous-time Riccati equation discrete-time Riccati equation continuous-time Lyapunov equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 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. 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. 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. 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.Google Scholar
  6. 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. 7.
    Potter, J. E.,Matrix Quadratic Solution, SIAM Journal on Applied Mathematics, Vol. 14, No. 3, pp. 496–501, 1964.Google Scholar
  8. 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.Google Scholar
  9. 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. 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. 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. 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. 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. 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. 15.
    Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Halsted Press, Washington, DC, p. 47, 1975.Google Scholar
  16. 16.
    Sage, A. P.,Optimum Systems Control, Englewood Cliffs, New Jersey, p. 131, 1968.Google Scholar
  17. 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.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • H. M. Chun
    • 1
  • J. D. Turner
    • 1
  1. 1.Control Theory/Dynamics GroupCambridge Research Division of Photon Research AssociatesCambridge

Personalised recommendations