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Journal of Optimization Theory and Applications

, Volume 111, Issue 2, pp 273–303 | Cite as

Multi-Input Multi-Output Ellipsoidal State Bounding

  • C. Durieu
  • É. Walter
  • B. Polyak
Article

Abstract

Ellipsoidal state outer bounding has been considered in the literature since the late sixties. As in the Kalman filtering, two basic steps are alternated: a prediction phase, based on the approximation of the sum of ellipsoids, and a correction phase, involving the approximation of the intersection of ellipsoids. The present paper considers the general case where K ellipsoids are involved at each step. Two measures of the size of an ellipsoid are employed to characterize uncertainty, namely, its volume and the sum of the squares of its semiaxes. In the case of multi-input multi-output state bounding, the algorithms presented lead to less pessimistic ellipsoids than the usual approaches incorporating ellipsoids one by one.

Bounded noise ellipsoidal bounding identification set-membership estimation state estimation 

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References

  1. 1.
    Witsenhausen, H. S., Sets of Possible States of Linear Systems Given Perturbed Observations, IEEE Transactions on Automatic Control, Vol. 13, pp. 556–558, 1968.Google Scholar
  2. 2.
    Schweppe, F. C., Recursive State Estimation: Unknown but Bounded Errors and System Inputs, IEEE Transactions on Automatic Control, Vol. 13, pp. 22–28, 1968.Google Scholar
  3. 3.
    Bertsekas, D. P., and Rhodes, I. B., Recursive State Estimation for a Set-Membership Description of Uncertainty, IEEE Transactions on Automatic Control, Vol. 16, pp. 117–128, 1971.Google Scholar
  4. 4.
    Kurzhanski, A. B., Control and Observation under Uncertainty, Nauka, Moscow, 1977 (in Russian).Google Scholar
  5. 5.
    Chernousko, F. L., State Estimation for Dynamic Systems, CRC Press, Boca Raton, Florida, 1994.Google Scholar
  6. 6.
    Kurzhanski, A. B., and Valyi, I., Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, Massachusetts, 1996.Google Scholar
  7. 7.
    Polyak, B. T., Convexity of Quadratic Transformations and Its Use in Control and Optimization, Journal of Optimization Theory and Applications, Vol. 99, pp. 553–583, 1998.Google Scholar
  8. 8.
    Chernousko, F. L., and Rokityanskii, D. Ya., Ellipsoidal Bounds on Reachable Sets of Dynamical Systems with Matrices Subjected to Uncertain Perturbations, Journal of Optimization Theory and Applications, Vol. 104, pp. 1–19, 2000.Google Scholar
  9. 9.
    Combettes, P. L., The Foundations of Set-Theoretic Estimation, Proceedings of the IEEE, Vol. 81, pp. 182–208, 1993.Google Scholar
  10. 10.
    Deller, J. R., Nayeri, M., and Odeh, S. F., Least-Square Identification with Error Bounds for Real-Time Signal Processing and Control, Proceedings of the IEEE, Vol. 81, pp. 813–849, 1993.Google Scholar
  11. 11.
    Walter, E., Editor, Special Issue on Parameter Identifications with Error Bound, Mathematics and Computers in Simulation, Vol. 32, pp. 447–607, 1990.Google Scholar
  12. 12.
    Norton, J. P., Editor, Special Issue on Bounded-Error Estimation, Part 1, International Journal of Adaptative Control and Signal Processing, Vol. 8, pp. 1–118, 1994.Google Scholar
  13. 13.
    Norton, J. P., Editor, Special Issue on Bounded-Error Estimation, Part 2, International Journal of Adaptative Control and Signal Processing, Vol. 9, pp. 1–132, 1995.Google Scholar
  14. 14.
    Milanese, M., Norton, J., Piet-Lahanier, H., and Walter, E., Editors, Bounding Approaches to System Identification, Plenum Press, New York, NY, 1996.Google Scholar
  15. 15.
    Walter, E., and Piet-Lahanier, H., Exact Recursive Polyhedral Description of the Feasible Parameter Set for Bounded-Error Models, IEEE Transactions on Automatic Control, Vol. 34, pp. 911–915, 1989.Google Scholar
  16. 16.
    Belforte, G., Bona, B., and Cerone, V., Parameter Estimation Algorithms for a Set-Membership Description of Uncertainty, Automatica, Vol. 26, pp. 887–898, 1990.Google Scholar
  17. 17.
    Fogel, E., and Huang, Y. F., On the Value of Information in System Identifi-cation: Bounded Noise Case, Automatica, Vol. 18, pp. 229–238, 1982.Google Scholar
  18. 18.
    Maksarov, D. G., and Norton, J. P., State Bounding with Ellipsoidal Set Description of the Uncertainty, International Journal of Control, Vol. 65, pp. 847–866, 1996.Google Scholar
  19. 19.
    Gollamudi, S., Nagaraj, S., Kapoor, S., and Huang, Y. F., Set-Membership State Estimation with Optimal Bounding Ellipsoids, Proceedings of the International Symposium on Information Theory and Its Applications, pp. 262–265, 1996.Google Scholar
  20. 20.
    Durieu, C., Polyak, B., and Walter, E., Trace Versus Determinant in Ellipsoidal Outer Bounding with Application to State Estimation, 13th IFAC World Congress, Vol. I, pp. 43–48, 1996.Google Scholar
  21. 21.
    Durieu, C., Polyak, B., and Walter, E., Ellipsoidal State Outer Bounding for MIMO Systems via Analytical Techniques, 1996 CESA IMACS IEEE-SMC Multiconference, Symposium on Modelling, Analysis, and Simulation, Vol. 2, pp. 843–848, 1996.Google Scholar
  22. 22.
    Durieu, C., Walter, E., and Polyak, B., Ellipsoidal Bounding Techniques for Parameter Tracking, 11th IFAC Symposium on System Identification, SYSID'97, Vol. 4, pp. 1777–1782, 1997.Google Scholar
  23. 23.
    Schweppe, F. C., Uncertain Dynamic Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1973.Google Scholar
  24. 24.
    Kahan, W., Circumscribing an Ellipsoid about the Intersection of Two Ellipsoids, Canadian Mathematical Bulletin, Vol. 2, pp. 437–441, 1968.Google Scholar
  25. 25.
    Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, Vol. 15, 1994.Google Scholar
  26. 26.
    Nesterov, Y., and Nemirovskii, A., Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, Vol. 13, 1994.Google Scholar
  27. 27.
    Horn, R. A., and Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, England, 1985.Google Scholar
  28. 28.
    Berger, M., Géométrie, CEDIC/Nathan, Paris, France, 1979.Google Scholar
  29. 29.
    Kiselev, O. N., and Polyak, B. T., Ellipsoidal Estimation with Respect to a Generalized Criterion, Automation and Remote Control, Vol. 52, pp. 1281–1292, 1992 (English Translation).Google Scholar
  30. 30.
    Kreindler, E., and Jameson, A., Conditions for Nonnegativeness of Partioned Matrices, IEEE Transactions on Automatic Control, Vol. 17, pp. 147–148, 1972.Google Scholar
  31. 31.
    Reshetnyak, Y., Summation of Ellipsoids in a Guaranteed Estimation Problem, Journal of Applied Mathematics and Mechanics (PMM), Vol. 53, pp. 249–254, 1989.Google Scholar
  32. 32.
    Bertsekas, D. P., Nonlinear Programming, Athena Publishers, Belmont, Massachusetts, 1998.Google Scholar
  33. 33.
    Pronzato, L., Walter, E., and Piet-Lahanier, H., Mathematical Equivalence of Two Ellipsoidal Algorithms for Bounded-Error Estimation, 28th IEEE Conference on Decision and Control, pp. 1952–1955, 1989.Google Scholar
  34. 34.
    Durieu, C., Walter, E., and Polyak, B., Set-Membership Estimation with the Trace Criterion Made Simpler Than with the Determinant Criterion, 12th IFAC Symposium on System Identification, SYSID'2000, CD-ROM, 2000.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • C. Durieu
    • 1
  • É. Walter
    • 2
  • B. Polyak
    • 3
  1. 1.Laboratoire d'Électricité, Signaux et RobotiqueCentre National de la Recherche Scientifique and École Normale Supérieure de CachanCachanFrance
  2. 2.Laboratoire des Signaux et Systemes, Centre National de la Recherche ScientifiqueÉcole Supérieure d'Électricité, and Université de Paris-SudGif-sur-YvetteFrance
  3. 3.Institute of Control ScienceRussian Academy of SciencesMoscowRussia

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