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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 869–896 | Cite as

Adaptive estimation for nonlinear systems using reproducing kernel Hilbert spaces

  • Parag BobadeEmail author
  • Suprotim Majumdar
  • Savio Pereira
  • Andrew J. Kurdila
  • John B. Ferris
Article

Abstract

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown or uncertain nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. Under the assumption that full state measurements are available, this paper (1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, (2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and (3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis.

Keywords

Adaptive estimation Reproducing kernel Hilbert spaces Distributed parameter systems 

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References

  1. 1.
    Wendland, H.: Scattered data approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  2. 2.
    Baumeister, J., Scondo, W., Demetriou, M.A., Rosen, I.G.: On-line parameter estimation for infinite dimensional dynamical systems. SIAM Journal of Control and Optimisation (1997)Google Scholar
  3. 3.
    Bohm, M., Demetriou, M.A., Reich, S., Rosen, I.G.: Model reference adaptive control of distributed parameter systems. SIAM Journal of Control and Optimisation (1998)Google Scholar
  4. 4.
    Chen Y., Dong, W., Zhao, Y., Farrell, J.A.: Tracking control for nonaffine systems: A self-organizing approximation approach. IEEE Transactions on Neural Networks and Learning Systems (2012)Google Scholar
  5. 5.
    Thrun, S., Burgard, W., Fox, D.: Probabilistic robotics (2005)Google Scholar
  6. 6.
    Durrant-Whyte, H., Bailey, T.: Simultaneous localization and mapping: part I. IEEE Robotics Automation Magazine (2006)Google Scholar
  7. 7.
    Bailey, T., Durrant-Whyte, H.: Simultaneous localization and mapping (SLAM): part II. IEEE Robotics Automation Magazine (2006)Google Scholar
  8. 8.
    Dissanayake, G., Huang, S., Wang, Z., Ranasinghe, R.: A review of recent developments in simultaneous localization and mapping. 6th International Conference on Industrial and Information Systems (2011)Google Scholar
  9. 9.
    Dissanayake, G., Durrant-Whyte, H., Bailey, T.: A computationally efficient solution to the simultaneous localisation and map building problem. In: Proceedings 2000 ICRA Millennium Conference (2000)Google Scholar
  10. 10.
    Huang, S., Dissanayake, G.: Convergence and consistency analysis for extended Kalman filter based SLAM. Trans. Robot. 23(5), 1036–1049 (2007)CrossRefGoogle Scholar
  11. 11.
    Julier, S.J., Uhlmann, J.K.: A counter example to the theory of simultaneous localization and map building. In: Proceedings 2001 IEEE International Conference on Robotics and Automation (2001)Google Scholar
  12. 12.
    Meyer, Y.: Wavelets and operators. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  13. 13.
    Mallat, S.: A wavelet tour of signal processing. Academic Press, New York (1999)zbMATHGoogle Scholar
  14. 14.
    Daubechies, I.: Ten lectures on wavelets. SIAM, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  15. 15.
    DeVore, R., Lorentz, G.: Constructive approximation (1993)Google Scholar
  16. 16.
    Opfer, R.: Tight frame expansions of multiscale reproducing kernels in Sobolev spaces. Applied Computational Harmonic Analysis (2006)Google Scholar
  17. 17.
    Opfer, R.: Multiscale kernels. Advances in Computational Mathematics (2006)Google Scholar
  18. 18.
    DeVore, R., Kerkyacharian, G., Picard, D., Temlyakov, V.: Approximation methods for supervised learning. Foundations of Computational Mathematics (2006)Google Scholar
  19. 19.
    Konyagin, S.V., Temlyakov, V.N.: The entropy in learning theory. Error estimates. Constr. Approx. 25(1), 1–27 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cohen, A., DeVore, R., Kerkyacharian, G., Picard, D.: Maximal spaces with given rate of convergence for thresholding algorithms. Applied and Computational Harmonic Analysis (2001)Google Scholar
  21. 21.
    Temlyakov, V.N.: Approximation in learning theory. Constructive Approximation (2008)Google Scholar
  22. 22.
    Sastry, S., Bodson, M.: Adaptive control: stability, convergence and robustness. Dover, New York (2011)zbMATHGoogle Scholar
  23. 23.
    Ioannou, P.A., Sun, J.: Robust adaptive control. Dover, New York (2012)zbMATHGoogle Scholar
  24. 24.
    Farrell, J.A., Polycarpou, M.M.: Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches. Wiley, New York (2006)CrossRefGoogle Scholar
  25. 25.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Adaptive boundary and point control of linear stochastic distributed parameter systems. SIAM J Control Optim. (1997)Google Scholar
  26. 26.
    Duncan, T.E., Pasik-Duncan, B.: Adaptive control of linear delay time systems. Stochastics (1988)Google Scholar
  27. 27.
    Duncan, T.E., Pasik-Duncan, B., Goldys, B.: Adaptive control of linear stochastic evolution systems. Stochastics Rep. (1991)Google Scholar
  28. 28.
    Pasik-Duncan, B.: On the consistency of a least squares identification procedure in linear evolution systems. Stochastics Report. (1992)Google Scholar
  29. 29.
    Hovakimyan, N., Cao, C.: \(\mathcal {L}^{1}\) Adaptive control theory. SIAM (2010)Google Scholar
  30. 30.
    Narendra, K., Annaswamy, A.M.: Stable adaptive systems. Prentice Hall, Upper Saddle River (1989)zbMATHGoogle Scholar
  31. 31.
    Narendra, K., Parthasarthy, K.: Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Networks (1990)Google Scholar
  32. 32.
    Narendra, K.S., Kudva, P.: Stable adaptive schemes for system identification and control - part II. IEEE Trans. Syst. Man Cybern., SMC- 4(6), 552–560 (1974)CrossRefzbMATHGoogle Scholar
  33. 33.
    Morgan, A.P., Narendra, K.S.: On stability of nonautonomous differential equations \(\dot {x}=[a+b(t)]x\), with skew symmetric b(t). SIAM Journal of Control and Optimisation (1977)Google Scholar
  34. 34.
    Banks, H.T., Kunisch, K.: Estimation techniques for distributed parameter systems. Birkhauser, Cambridge (1989)CrossRefzbMATHGoogle Scholar
  35. 35.
    Smale, S., Zhou, X.: Learning theory estimates via integral operators and their approximations. Constructive Approximation (2007)Google Scholar
  36. 36.
    DeVore, R.: Adapting to unknown smoothness via wavelet shrinkage. Acta Numerica (1998)Google Scholar
  37. 37.
    Adams, R.A., Fournier, John: Sobolev spaces. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  38. 38.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (2011)zbMATHGoogle Scholar
  39. 39.
    Farkas, B., Wegner, S.-A.: Variations on Barbalat’s lemma. arXiv:1411.1611v3 (2016)
  40. 40.
    Demetriou, M.A., Rosen, I.G.: On the persistence of excitation in the adaptive identification of distributed parameter systems. IEEE Transactions of Automatic Control (1994)Google Scholar
  41. 41.
    Narendra, K.S., Annaswamy, A.M.: Persistent excitation in adaptive systems. International Journal of Control (1987)Google Scholar
  42. 42.
    Demetriou, M.A.: Adaptive parameter estimation of abstract parabolic and hyperbolic distributed parameter systems. Phd thesis, University of Southern California, Los Angeles (1993)Google Scholar
  43. 43.
    Demetriou, M.A., Rosen, I.G.: Adaptive identification of second order distributed parameter systems. Inverse Problems (1994)Google Scholar
  44. 44.
    Kazimir, Joseph, Rosen, I.G.: Adaptive estimation of nonlinear distributed parameter systems. International Series of Numerical Mathematics. Birkhauser Verlag, Basel (1994)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Science and MechanicsVirginia TechBlacksburgUSA
  2. 2.Department of Electrical and Computer EngineeringVirginia TechBlacksburgUSA
  3. 3.Department of Mechanical EngineeringVirginia TechBlacksburgUSA

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