On Stability of Guaranteed Estimation Problems: Error Bounds for Information Domains and Experimental Design

  • Mikhail I. Gusev
  • Sergei A. Romanov
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 57)

Abstract

The two guaranteed estimation problems related to semi-infinite programming theory are considered. For abstract estimation problem in a Banach space we study the dependence of information domains ([12], [14]) on measurement errors with intensity of errors tending to zero. The upper estimates for the rate of convergence of information domains to their limit in the Hausdorff metric are given. The experimental design problem for estimation of distributed system with uncertain parameters through available measurements is also considered in the context of guaranteed estimation theory. For the stationary sensor placement problem we describe its reduction to a nonlinear programming problem. In the case of sufficiently large number of sensors it is shown that the solution may be obtained by solving linear semi-infinite programming problem.

Keywords

Optimal Control Problem Uncertain Parameter Uncertain System Linear Continuous Operator Sensor Placement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.A. Anikin. Error estimate for a regularization method in problems of the reconstruction of inputs os dynamic systems, Computational Mathematics and Mathematical Physics, 37:1056–1067, 1997.MathSciNetMATHGoogle Scholar
  2. [2]
    J.P. Aubin and I. Ekeland. Applied Nonlinear Analysis, Wiley, 1984.MATHGoogle Scholar
  3. [3]
    W.H. Chen and J.H. Seinfeld. Optimal allocation of process measurements, International Journal of Control, 21:1003–1014, 1975.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M.I. Gusev. On the stability of solution of the inverse problems in control system dynamics, Problems of Control and Information Theory, 17:297– 310, 1988.MathSciNetMATHGoogle Scholar
  5. [5]
    M.I. Gusev. Measurement allocation problems in estimation of dynamical systems under geometrical constraints on uncertainty, Diff: Uravn., 24:1862–1870, (in Russian) 1988.MathSciNetGoogle Scholar
  6. [6]
    M.I. Gusev. On the optimality of linear algorithms in guaranteed estimation. In A.B. Kurzhanski and V.M. Veliov, editors, Modeling Techniques for Uncertain Systems, Birkhäuser, Boston, pages 93–110, 1994.Google Scholar
  7. [7]
    M.I. Gusev. On Stability of Information Domains in Guaranteed Estimation Problem. In Proceedings of the Steklov Institute of Mathematics, Suppl. 1, MAIK “Nauka/Interperiodika”, pages 104–118, 2000.Google Scholar
  8. [8]
    A.J. Hoffman. On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, 49:263–265, 1952.MathSciNetCrossRefGoogle Scholar
  9. [9]
    V.K. Ivanov, V.V. Vasin and V.P. Tanana. Theory of Linear Ill-Posed Prolems and Applications, Nauka, Moscow, (in Russian) 1978 .Google Scholar
  10. [10]
    D. Klatte and W. Li. Asymptotic constraint qualification and global error bounds for convex inequalities, Mathematical Programming, 8A:137– 160, 1999.MathSciNetGoogle Scholar
  11. [11]
    E. K. Kostousova. Approximation of the problem of choosing an optimal composition of measurements in a parabolic system, U. S. S.R. Computational Mathematics and Mathematical Physics, 30: 8–17, 1990.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    A. B. Kurzhanski. Control and Observation under Uncertainty Conditions, Nauka, Moscow, (in Russian) 1977.Google Scholar
  13. [13]
    A. B. Kurzhanski and A.Yu. Khapalov. Estimation of distributed fields according to results of observations. In Partial Differential Equations Proceedings of the International Conference in Novosibirsk 1983, pages 102–108, (in Russian) 1986.Google Scholar
  14. [14]
    A. B. Kurzhanski and I. Valyi. Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997.MATHCrossRefGoogle Scholar
  15. [15]
    A.B. Kurzhanski and I.F. Sivergina. On noninvertible evolutionary systems: guaranteed estimation and regularization problems, Dokl. Acad. Nauk SSSR, 314:292–296(in Russian) , 1990.Google Scholar
  16. [16]
    A.B. Kurzhanski and I.F. Sivergina. Method of guaranteed estimates and regularization problems for evolutionary system, Journal vichislit. rnatem. i mat. phiziki., 32:1720–1733(in Russian) , 1992.Google Scholar
  17. [17]
    G.I. Marchuk, Mathematical Modelling and Environmental Problems. Nauka, Moscow, (in Russian) 1982.Google Scholar
  18. [18]
    M. Milanese and J. Norton (editors). Bounding Approaches to System Identification, Plenum Press, London, 1996.MATHGoogle Scholar
  19. [19]
    J. Nakamori, S. Miyamoto, S. Ikeda and J. Savaragi. Measurement optimization with sensitivity criteria for distributed parameter systems, IEEE Transactions Automatic Control, AC-25:889–900, 1980.MATHCrossRefGoogle Scholar
  20. [20]
    L. W. Neustadt. Optimization, a moment problem, and nonlinear programming, SIAM Journal of Control, Ser. A, 2:33–53, 1964.MathSciNetMATHGoogle Scholar
  21. [21]
    V. Pokotilo. Necessary Conditions for Measurements Optimization. In A.B Kurzhanski and V.M. Veliov, editors, Modeling Techniques for Uncertain Systems, Birkhäuser, Boston, pages 147–159, 1994.Google Scholar
  22. [22]
    E. Rafajlowicz. Design of experiments for eigenvalue in identification in distributed parameter system, International Journal of Control, 34:1079–1094, 1981.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    S.M. Robinson. An application of error bounds for convex programming in a linear space, SIAM Journal on Control and Optimmization, 13:271–273, 1975.MATHCrossRefGoogle Scholar
  24. [24]
    A.N. Tikhonov and V.Ya. Arsenin. Methods of Solutions of Ill-Posed Problems, Nauka, Moscow, (in Russian) 1974.Google Scholar
  25. [25]
    D. Ucinski. Measurement Optimization for Parameter Estimation in Distributed Systems, Technical University Press, Zielona Gora, Poland, 1999.Google Scholar
  26. [26]
    D.J. Ucinski, J. Korbicz and M. Zaremba. On optimization of sensors motions in parameter identification of two-dimensional distributed systems, In Proc. 2nd European Control Con ference, pages 1359–1364, Groningen, The Netherlands, 1992.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Mikhail I. Gusev
    • 1
  • Sergei A. Romanov
    • 1
  1. 1.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesEkaterinburgRussia

Personalised recommendations