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Authors
R. McVinish; J. H. Braslavsky; K. Mengersen

Identifier
10.3182/20060329-3-AU-2901.00162

Index Terms
transfer functions,decision theory,non-parametric identification,Monte Carlo calculation,loss minimization

Abstract
This paper takes a Bayesian-decision theoretic approach to transfer function estimation, nominal model estimation, and quantification of the resulting model error. Consistency of the nonparametric estimate of the transfer function is proved together with a rate of convergence. The required quantities can be computed routinely using reversible jump Markov chain Monte Carlo methods. The proposed methodology has connections with set membership identification which has been extensively studied for this problem.

References

[1] Akçay, H., H. Hjalmarsson and L. Ljung (1996). On
    the choice of norms in system identification.
    IEEE Trans. on Automatic Control 41, 1367-
    1372.
[2] Berger, J.O. (1985). Statistical Decision Theory and
    Bayesian Analysis, 2nd Ed., Springer-Verlag.
    New York.
[3] Brooks, S. P., P. Guidici and G. O. Roberts (2003).
    Efficient construction of reversible jump Markov
    chain Monte Carlo proposal distributions. J. R.
    Statist. Soc. B. 65, 3-55.
[4] Garulli, A., A. Vicino and G. Zappa (2000). Conditional
     central algorithms for worst case set-membership
     identification and filtering. IEEE.
    Trans. on Automatic Control 45, 14-23.
[5] Ghosal, S. and A. W. van der Vaart (2005). Convergence
     rates of posterior distributions for non iid
    observations. Ann. Statist. (accepted).
[6] Ghosh, J. K. and R. V. Ramamoorthi (2003). Bayesian
    nonparametrics. Springer. New York.
[7] Green, P. (1995). Reversible jump MCMC computation
     and Bayesian model determination.
    Biometrika 82, 711-732.
[8] Hildebrand, R. and M. Gevers (2003). Identification
    for control: optimal input design with respect to
    a worst-case ν-gap cost function. SIAM J. Control
    Optim. 41, 1586-1608.
[9] Hjalmarsson, H. (2005). From experiment design to
    closed-loop control. Automatica 41, 393-438.
[10] Juloski, A. L., S. Weiland and W. P. Heemels (2004).
     A Bayesian approach to identification of hybrid
     systems. In: Proceedings of the 43rd IEEE Conference
      on Decision and Control. Atlantis, Paradise
      Island, Bahamas. pp. 13-19.
[11] Milanese, M. and A. Vicino (1991). Optimal estimation
      theory for dynamic systems with set membership
      uncertainty: An overview. Automatica
     27, 997-1009.
[12] Muller, P. and F. A. Quintana (2004). Nonparametric
      Bayesian data analysis. Statistical Science
     19, 95-110.
[13] Ninness, B., S. Henriksen and T. Brinsmead (2002).
     System identification via a computational
     Bayesian approach. In: Proceedings of the 41st
     IEEE Conference on Decision and Control. Las
     Vegas, Nevada, USA. pp. 1820-1825.
[14] Reinelt, W., A. Garulli and L. Ljung (2002). Comparing
      different approaches to model error modeling
     in robust identification. Automatica 38, 787-803.
[15] Robert, C. P. and G. Casella (2004). Monte Carlo statistical
      methods. Springer. New York.
[16] Šmídl, V., A. Quinn, M. Kárny and T. V. Guy (2005).
     Robust estimation of autoregressive processes using
      a mixture-based filter-bank. Systems & Control
      Letters 54, 315-323.