Skip to main content
SpringerLink
Log in

Linear quadratic optimal control for discrete descriptor systems

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The present paper deals with the investigation of the linear quadratic optimal control problem for the discrete-time descriptor systemsEx k+1=Ax k +Bu k , whereE is in general a singular matrix and the system structure is in general noncausal. The problem is considered in its general form, having singular cost matrices and cross-weighting term in the cost functional. The key idea for the solution approach is the use of the Weierstrass theorem for regular pencils, combined with a suitable permutation transformation, to form a base for the image ofE. The optimization problem is solved by forcing causality to the Hamiltonian equations, which are produced by considering the entireN-stage process as a large system of linear equations. The feedback gain matrix is obtained as a manifold which is generated by the intersection of two other manifolds.

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. Bellman, R.,Introduction to the Mathematical Theory of Control Processes, Academic Press, New York, New York, 1967.

    Google Scholar 

  2. Bellman, R., andDreyfus, S.,Applied Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1962.

    Google Scholar 

  3. Bellman, R.,Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.

    Google Scholar 

  4. Silverman, L. M.,Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties, and System Theory Interpretations, Control and Dynamic Systems, Edited by C. T. Leondes, Academic Press, New York, New York, Vol. 12, pp. 313–386, 1967.

    Google Scholar 

  5. Kalman, R. E., andEnglar, T. S.,A User's Manual for ASP, NASA, Conference Report No. CR-475, 1966.

  6. Payne, H. J., andSilverman, L. M.,On the Discrete-Time Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. AC-18, pp. 226–234, 1973.

    Google Scholar 

  7. Bell, D. J., andJacobson, D. H.,Singular Optimal Control Problems, Academic Press, New York, New York, 1975.

    Google Scholar 

  8. Anderson, B. D. O., andMoore, J. B.,Optimal Filtering, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.

    Google Scholar 

  9. Campbell, S. L.,Optimal Control of Discrete Linear Processes with Quadratic Cost, International Journal of Systems Science, Vol. 9, pp. 841–847, 1978.

    Google Scholar 

  10. Lewis, F. L.,A Generalized Inverse Solution to the Discrete-Time Singular Riccati Equation, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 395–398, 1981.

    Google Scholar 

  11. Rosenbrock, H. H.,State-Space and Multivariable Theory, Nelson, London, England, 1970.

    Google Scholar 

  12. Rosenbrock, H. H., andPugh, A. C.,Contributions to a Hierarchical Theory of Systems, International Journal of Control, Vol. 19, pp. 845–867, 1974.

    Google Scholar 

  13. Leontief, W.,Essays in Economics, M. E. Sharpe, New York, New York, 1977.

    Google Scholar 

  14. Luenberger, D. G.,Dynamic Equations in Descriptor Form, IEEE Transactions on Automatic Control, Vol. AC-22, pp. 312–321, 1977.

    Google Scholar 

  15. Crandal, S. H., Karnopp, D. C., Kurtz, E. F., andPridmore-Brown, D. C.,Dynamics of Mechanical and Electromechanical Systems, McGraw-Hill, New York, New York, 1968.

    Google Scholar 

  16. Anderson, D. J.,Matrix Methods for the Study of a Regulated Synchronous Machine, Proceedings of the IEEE, Vol. 57, pp. 2122–2136, 1969.

    Google Scholar 

  17. Kokotovic, P. V., andYackel, R. A.,Singular Perturbation of Linear Regulators: Basic Theorems, IEEE Transactions on Automatic Control, Vol. AC-17, pp. 29–37, 1972.

    Google Scholar 

  18. Rosenbrock, H. H.,Structural Properties of Linear Dynamical Systems, International Journal of Control, Vol. 20, pp. 191–202, 1974.

    Google Scholar 

  19. Luenberger, D. G.,Time Invariant Descriptor Systems, Automatica, Vol. 14, pp. 473–480, 1978.

    Google Scholar 

  20. Luenberger, D. G.,Time Invariant Descriptor Systems, Proceedings of the Joint Automatic Control Conference, San Francisco, California, pp. 725–750, 1977.

  21. Luenberger, D. G., andArbel, A.,Singular Dynamic Leontief Systems, Econometrica, Vol. 45, pp. 991–995, 1977.

    Google Scholar 

  22. Yamada, T., andLuenberger, D. G.,Generic Controllability Theorems for Descriptor Systems, IEEE Transactions on Automatic Control, Vol. AC-30, pp. 144–152, 1985.

    Google Scholar 

  23. Campbell, S. L.,Singular Systems of Differential Equations, Vols. 1–2, Pitman, London, England, 1980–1982.

    Google Scholar 

  24. Verghese, G.,Infinite-Behavior in Generalized Dynamical Systems, PhD Thesis, Stanford University, Palo Alto, California, 1979.

    Google Scholar 

  25. Verghese, G., Levy, B., andKailath, T.,A Generalized State Space for Singular Systems, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 811–830, 1981.

    Google Scholar 

  26. Pandolfi, L.,On the Regulator Problem for Linear Degenerate Control Systems, Journal of Optimization Theory and Applications, Vol. 33, pp. 241–254, 1981.

    Google Scholar 

  27. Cobb, D.,Feedback and Pole Placement in Descriptor-Variable Systems, International Journal of Control, Vol. 33, pp. 1135–1146, 1981.

    Google Scholar 

  28. Cobb, D.,Descriptor-Variable Systems and Optimal State Regulation, IEEE Transactions on Automatic Control, Vol. AC-28, pp. 601–611, 1983.

    Google Scholar 

  29. Cobb, D.,Controllability, Observability, and Duality in Singular Systems, IEEE Transactions on Automatic Control, Vol. AC-29, pp. 1076–1082, 1984.

    Google Scholar 

  30. Larson, R. E., Luenberger, D. G., andStengel, D. N.,Descriptor Variable Theory and Spatial Dynamic Programming, Systems Control, SCI Report No. TM-5168-9, 1978.

  31. Yip, E. L., andSincovec, R. F.,Solvability, Controllability, and Observability of Continuous Descriptor Systems, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 702–707, 1981.

    Google Scholar 

  32. Sincovec, R. F., Erisman, A. M., Yip, E. L., andEpton, M. A.,Analysis of Descriptor Systems Using Numerical Algorithms, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 139–147, 1981.

    Google Scholar 

  33. Gantmacher, F. R.,The Theory of Matrices, Vol. 2, Chelsea, New York, New York, 1974.

    Google Scholar 

  34. Langenhop, C. E.,Laurent Expansion for a Nearly Singular Matrix, Linear Algebra and Its Applications, Vol. 4, pp. 329–340, 1971.

    Google Scholar 

  35. Kai-Tak Wong,The Eigenvalue Problem λTx+Sx, Journal of Differential Equations, Vol. 16, pp. 270–280, 1974.

    Google Scholar 

  36. Van Dooren, P.,A Generalized Eigenvalue Approach for Solving Riccati Equations, SIAM Journal on Scientific and Statistical Computing, Vol. 2, pp. 121–135, 1981.

    Google Scholar 

  37. Arnold, W. F., andLaub, A. J.,Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations, Proceedings of the IEEE, Vol. 72, pp. 1746–1754, 1984.

    Google Scholar 

  38. Gaalman, G. J.,Comments on “A Nonrecursive Algebraic Solution for the Discrete Riccati Equation IEEE Transactions on Automatic Control, Vol. AC-25, pp. 610–612, 1980.

    Google Scholar 

  39. Pappas, T., Laub, A. J., andSandell, N. R.,On the Numerical Solution of the Discrete-Time Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 631–641, 1980.

    Google Scholar 

  40. Ben-Israel, A., andGreville, T. N. E.,Generalized Inverses: Theory and Applications, Krieger Publishing Company, Huntington, New York, 1980.

    Google Scholar 

  41. Lewis, F. L., andOzcaldiran, K.,Reachability and Controllability for Descriptor Systems, Proceedings of the 27th Midwest Symposium on Circuits and Systems, Morgantown, West Virginia, 1984.

  42. Mantas, G. P.,Optimal Control of Descriptor Systems, National Technical University of Athens, Doctoral Thesis, 1987.

Download references

Author information

Authors and Affiliations

  1. Control Systems Division, National Technical University of Athens, Athens, Greece

    G. P. Mantas (Research Assistant) & N. J. Krikelis (Professor)

Authors
  1. G. P. Mantas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. J. Krikelis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by C. T. Leondes

This work was supported in part by a Research Fund from the National Technical University of Athens, Athens, Greece.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mantas, G.P., Krikelis, N.J. Linear quadratic optimal control for discrete descriptor systems. J Optim Theory Appl 61, 221–245 (1989). https://doi.org/10.1007/BF00962798

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00962798

Key Words

Search

Navigation