Guaranteed State Estimation

Part of the Systems & Control: Foundations & Applications book series (SCFA, volume 85)


This chapter deals with the problem of set-membership or “guaranteed” state estimation. The problem is to estimate the state of a dynamic process from partial observations corrupted by unknown but bounded noise in the system and measurement inputs (in contrast with stochastic noise). The problem is treated in both continuous and discrete time. Comparison with stochastic filtering is also discussed.


State estimation Bounding approach Information states Information tubes Hamiltonian techniques Ellipsoidal approximations 


  1. 14.
    Baras, J.S., Kurzhanski, A.B.: Nonlinear filtering: the set-membership (bounding) and the H techniques. In: Proceedings of the 3rd IFAC Symposium NOLCOS, pp. 409–418. Pergamon Press, Oxford (1995)Google Scholar
  2. 16.
    Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. SCFA. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  3. 17.
    Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton–Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1742 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 25.
    Bertsekas, D.P., Rhodes, I.: Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Automat. Control 16(2), 117–128 (1971)MathSciNetCrossRefGoogle Scholar
  5. 45.
    Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)Google Scholar
  6. 50.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–41 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 56.
    Daryin, A.N., Digailova, I.A., Kurzhanski, A.B.: On the problem of impulse measurement feedback control. Proc. Steklov Math. Inst. 268(1), 71–84 (2010)MathSciNetCrossRefGoogle Scholar
  8. 80.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)zbMATHGoogle Scholar
  9. 108.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME 82(Series D), 35–45 (1960)Google Scholar
  10. 110.
    Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Courier Dover, New York (1999)Google Scholar
  11. 113.
    Koscheev, A.S., Kurzhanski, A.B.: On adaptive estimation of multistage systems under uncertainty. Izvestia Academy Sci. 2, 72–93 (1983)Google Scholar
  12. 119.
    Krasovski, N.N.: On the theory of controllability and observability of linear dynamic systems. Appl. Math. Mech. (PMM) 28(1), 3–14 (1964)Google Scholar
  13. 135.
    Kurzhanski, A.B.: On duality of problems of optimal control and observation. Appl. Math. Mech. 34(3), 429–439 (1970) [translated from the Russian]MathSciNetGoogle Scholar
  14. 137.
    Kurzhanski, A.B.: Control and Observation Under Uncertainty Conditions, 392 pp. Nauka, Moscow (1977)Google Scholar
  15. 138.
    Kurzhanskii, A.B.: On stochastic filtering approximations of estimation problems for systems with uncertainty. Stochastics 23(6), 109–130 (1982)MathSciNetGoogle Scholar
  16. 140.
    Kurzhanski, A.B.: On the analytical description of the set of viable trajectories of a differential system. Dokl. Akad. Nauk SSSR. 287(5), 1047–1050 (1986)MathSciNetGoogle Scholar
  17. 158.
    Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes: a mathematical formalism for uncertain dynamics, viability and control. In: Advances in Nonlinear Dynamics and Control. Progress in Systems and Control Theory, vol. 17, pp. 122–188. Birkhäuser, Boston (1993)Google Scholar
  18. 159.
    Kurzhanski, A.B., Filippova, T.F., Sugimoto, K., Valyi, I.: Ellipsoidal state estimation for uncertain dynamical systems. In: Milanese, M., Norton, J., Piet-Lahanier, H., Walter, E. (eds.) Bounding Approaches to System Identification, pp. 213–238. Plenum Press, New York (1996)Google Scholar
  19. 167.
    Kurzhanski, A.B., Sivergina, I.F.: Method of guaranteed estimates and regularization problems for evolutionary systems. J. Comput. Math. Math. Phys. 32(11), 1720–1733 (1992)Google Scholar
  20. 168.
    Kurzhanski, A.B., Sugimoto, K., Vályi, I.: Guaranteed state estimation for dynamical systems: ellipsoidal techniques. Int. J. Adapt. Control Signal Process. 8(1), 85–101 (1994)CrossRefzbMATHGoogle Scholar
  21. 174.
    Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  22. 210.
    Milanese, M., Norton, J., Piet-Lahanier, H., Walter, E. (eds.): Bounding Approach to System Identification. Plenum Press, London (1996)Google Scholar
  23. 241.
    Schlaepfer, F.M., Schweppe, F.C.: Continuous-time state estimation under disturbances bounded by convex sets. IEEE Trans. Automat. Control 17(2), 197–205 (1972)CrossRefzbMATHGoogle Scholar
  24. 242.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1950)zbMATHGoogle Scholar
  25. 247.
    Subbotin, A.I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective. SCFA. Birkhäuser, Boston (1995)CrossRefGoogle Scholar
  26. 271.
    Witsenhausen, H.: Set of possible states of linear systems given perturbed observations. IEEE Trans. Automat. Control 13(1), 5–21 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State (Lomonosov) UniversityMoscowRussia
  2. 2.Electrical Engineering and Computer SciencesUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations