Abstract
We consider an estimation problem which appears in the identification of systems by means of restricted complexity models: find the optimal approximation to an element of a linear normed space (a system) based on noisy information, subject to the restriction that approximations (models) can be selected from a prescribed subspace M of the problem element space. In contrast to the worst-case optimization criterion, which may be pessimistic, in this paper the quality of an identification algorithm is measured by its local average performance. Two types of local average errors are considered: for a given information (measurement) y and for a given unknown element x, the latter in two versions. For a wide spectrum of norms in the measurement space, we define an optimal algorithm and give expressions for its average errors which show the dependence on information, information errors, unmodelled dynamics, and norm in the measurement space.
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Chen, J., Nett, C. N., and Fan, M.K., Optimal Nonparametric System Identification from Arbitrary Corrupt Finite Time Series: A Worst-Case/Deterministic Approach, Proceedings of the American Control Conference, Chicago, Illinois, 1992.
Chernousko, F. L., State Estimation for Dynamic Systems, CRC Press, Boca Raton, Florida, 1994.
Garulli, A., Kacewicz, B., Vicino, A., and Zappa, G., Error Bounds for Conditional Algorithms in Restricted Complexity Set Membership Identification, IEEE Transactions on Automatic Control, Vol. 45, pp. 160-164, 2000.
Garulli, A., Kacewicz, B., Vicino, A., and Zappa, G., Reliability of Projection Algorithms in Conditional Estimation, Journal of Optimization Theory and Applications, Vol. 101, pp. 1-14, 1999.
GiarrÉ, L., Kacewicz, B., and Milanese, M., Model Quality Evaluation in Set Membership Identification, Automatica, Vol. 33, pp. 1133-1140, 1997.
Harrison, K. J., Partington, J. R., and Ward, J. A., Complexity of Identification of Linear Systems with Rational Transfer Functions, Mathematics of Control Signals Systems, Vol. 11, pp. 265-288, 1998.
Helmicki, A. J., Jacobson, C. A., and Nett, C. N., Control Oriented System Identification: A Worst-Case/Deterministic Approach in H ∞ IEEE Transactions on Automatic Control, Vol. 36, pp. 1163-1176, 1991.
Kacewicz, B., Worst-Case Conditional System Identification in a General Class of Norms, Automatica, Vol. 35, pp. 1049-1058, 1999.
Kacewicz, B., Milanese, M., Tempo, R., and Vicino, A., Optimality of Central and Projection Algorithms for Bounded Uncertainty, Systems and Control Letters, Vol. 8, pp. 161-171, 1986.
Kacewicz, B., Milanese, M., and Vicino, A., Conditionally Optimal Algorithms and Estimation of Reduced-Order Models, Journal of Complexity, Vol. 4, pp. 73-85, 1988.
MÄkilÄ, P., and Partington, J., On Robustness in System Identification, Automatica, Vol. 35, pp. 907-916, 1999.
MÄkilÄ, P., Partington, J., and Gustafsson, T. K., Worst-Case Control-Relevant Identification, Automatica, Vol. 31, pp. 1799-1819, 1996.
Milanese, M., Norton, J., Piet-Lahanier, H., and Walter, E., Editors, Bounding Approaches to System Identification, Plenum Press, New York, NY, 1996.
Milanese, M., and Vicino, A., Information-Based Complexity and Nonparametric Worst-Case System Identification, Journal of Complexity, Vol. 9, pp. 427-445, 1993.
Ninness, B., and Goodwin, G., Estimation of Model Quality, Automatica, Vol. 31, pp. 1771-1798, 1996.
Traub, J., Wasilkowski, G., and WoŹniakowski, H., Information-Based Complexity, Academic Press, New York, NY, 1988.
Plaskota, L., Noisy Information and Computational Complexity, Cambridge University Press, Cambridge, England, 1996.
Kacewicz, B., Optimal Average Case Estimation in Hilbert Norms, Mathematics of Control Signals Systems, Vol. 13, pp. 347-359, 2000.
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Kacewicz, B. Optimal Conditional Estimation: Average Case Setting. Journal of Optimization Theory and Applications 109, 649–666 (2001). https://doi.org/10.1023/A:1017524023577
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DOI: https://doi.org/10.1023/A:1017524023577