Skip to main content
SpringerLink
Log in

Minimizing Sequences for a Family of Functional Optimal Estimation Problems

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

Abstract

Rates of convergence are derived for approximate solutions to optimization problems associated with the design of state estimators for nonlinear dynamic systems. Such problems consist in minimizing the functional given by the worst-case ratio between the ℒ p -norm of the estimation error and the sum of the ℒ p -norms of the disturbances acting on the dynamic system. The state estimator depends on an innovation function, which is searched for as a minimizer of the functional over a subset of a suitably-defined functional space. In general, no closed-form solutions are available for these optimization problems. Following the approach proposed in (Optim. Theory Appl. 134:445–466, 2007), suboptimal solutions are searched for over linear combinations of basis functions containing some parameters to be optimized. The accuracies of such suboptimal solutions are estimated in terms of the number of basis functions. The estimates hold for families of approximators used in applications, such as splines of suitable orders.

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. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  2. Alessandri, A., Sanguineti, M.: Input-output stability for optimal estimation problems. Int. Math. Forum 2, 593–617 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Alessandri, A., Cervellera, C., Sanguineti, M.: Design of asymptotic estimators: an approach based on neural networks and nonlinear programming. IEEE Trans. Neural Netw. 18, 86–96 (2007)

    Article  Google Scholar 

  4. Alessandri, A., Cervellera, C., Sanguineti, M.: Functional optimal estimation problems and their approximate solution. Optim. Theory Appl. 134, 445–466 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alessandri, A., Cervellera, C., Macciò, D., Sanguineti, M.: Optimization based on quasi-Monte Carlo sampling to design state estimators for nonlinear systems. Optimization (2010). doi:10.1080/02331930902863665

  6. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (2008)

    MATH  Google Scholar 

  7. Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930–945 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Daniel, J.W.: The Approximate Minimization of Functionals. Prentice-Hall, Englewood Cliffs (1971)

    MATH  Google Scholar 

  9. Gauthier, J.-P., Kupka, I.: Deterministic Observation Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  10. Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall, Englewood Cliffs (1963)

    Google Scholar 

  11. Hornik, K., Stinchcombe, M., White, H., Auer, P.: Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives. Neural Comput. 6, 1262–1275 (1994)

    Article  MATH  Google Scholar 

  12. Kůrková, V., Sanguineti, M.: Comparison of worst-case errors in linear and neural network approximation. IEEE Trans. Inf. Theory 48, 265–275 (2002)

    Google Scholar 

  13. Kůrková, V., Sanguineti, M.: Error estimates for approximate optimization by the extended Ritz method. SIAM J. Optim. 15, 261–287 (2005)

    Google Scholar 

  14. Kůrková, V., Sanguineti, M.: Geometric upper bounds on rates of variable-basis approximation. IEEE Trans. Inf. Theory 54, 5681–5688 (2008)

    Article  Google Scholar 

  15. Kainen, P.C., Kůrková, V., Sanguineti, M.: Minimization of error functionals over variable-basis functions. SIAM J. Optim. 14, 732–742 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kalsi, K., Lian, J., Hui, S., Zak, S.H.: Sliding-mode observers for systems with unknown inputs: a high-gain approach. Automatica 46, 347–353 (2010)

    Article  MATH  Google Scholar 

  17. Kleinman, D.L., Athans, M.: The design of suboptimal linear time-varying systems. IEEE Trans. Autom. Control 13, 150–159 (1968)

    Article  MathSciNet  Google Scholar 

  18. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  19. Wahba, G.: Spline Models for Observational Data. SIAM, Philadelphia (1990)

    MATH  Google Scholar 

  20. Zoppoli, R., Sanguineti, M., Parisini, M.: Approximating networks and extended Ritz method for the solution of functional optimization problems. J. Optim. Theory Appl. 112, 403–439 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Production Engineering, Thermoenergetics, and Mathematical Models (DIPTEM), University of Genoa, P. le Kennedy, Pad. D, 16129, Genova, Italy

    Angelo Alessandri

  2. Department of Communications, Computer, and System Sciences (DIST), University of Genoa, Via Opera Pia 13, 16145, Genova, Italy

    Giorgio Gnecco & Marcello Sanguineti

Authors
  1. Angelo Alessandri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Gnecco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcello Sanguineti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcello Sanguineti.

Additional information

Communicated by V.F. Demyanov.

G. Gnecco and M. Sanguineti were partially supported by the project Ateneo 2008 “Solution of functional optimization problems by nonlinear approximators and learning from data” of the University of Genoa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alessandri, A., Gnecco, G. & Sanguineti, M. Minimizing Sequences for a Family of Functional Optimal Estimation Problems. J Optim Theory Appl 147, 243–262 (2010). https://doi.org/10.1007/s10957-010-9720-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9720-3

Keywords

Search

Navigation