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Multi-Input Multi-Output Ellipsoidal State Bounding

Author

Listed:
  • C. Durieu

    (Centre National de la Recherche Scientifique and École Normale Supérieure de Cachan)

  • É. Walter

    (École Supérieure d'Électricité, and Université de Paris-Sud)

  • B. Polyak

    (Russian Academy of Sciences)

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.

Suggested Citation

  • C. Durieu & É. Walter & B. Polyak, 2001. "Multi-Input Multi-Output Ellipsoidal State Bounding," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 273-303, November.
    • Handle: RePEc:spr:joptap:v:111:y:2001:i:2:d:10.1023_a:1011978200643
    DOI: 10.1023/A:1011978200643
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    References listed on IDEAS

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    1. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, December.
    2. F. L. Chernousko & D. Ya. Rokityanskii, 2000. "Ellipsoidal Bounds on Reachable Sets of Dynamical Systems with Matrices Subjected to Uncertain Perturbations1," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 1-19, January.
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    Cited by:

    1. Ligang Sun & Hamza Alkhatib & Boris Kargoll & Vladik Kreinovich & Ingo Neumann, 2019. "Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems," Mathematics, MDPI, vol. 7(12), pages 1-22, December.
    2. Amirreza Fahim Golestaneh, 2022. "A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in R 3," Mathematics, MDPI, vol. 11(1), pages 1-21, December.
    3. Zhang, Liang & Feng, Zhiguang & Jiang, Zhengyi & Zhao, Ning & Yang, Yang, 2020. "Improved results on reachable set estimation of singular systems," Applied Mathematics and Computation, Elsevier, vol. 385(C).
    4. Liao, Wei & Liang, Taotao & Wei, Xiaohui & Yin, Qiaozhi, 2022. "Probabilistic reach-Avoid problems in nondeterministic systems with time-Varying targets and obstacles," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    5. Yongchun Jiang & Hongli Yang & Ivan Ganchev Ivanov, 2024. "Reachable Set Estimation and Controller Design for Linear Time-Delayed Control System with Disturbances," Mathematics, MDPI, vol. 12(2), pages 1-10, January.
    6. Nguyen D. That & Phan T. Nam & Q. P. Ha, 2013. "Reachable Set Bounding for Linear Discrete-Time Systems with Delays and Bounded Disturbances," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 96-107, April.

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