Abstract
The general schemes of linear estimation and filtration were considered on assumption of the unknown covariance matrix of random factors such as unknown parameters, measurement errors, and initial and external perturbations. A new criterion was introduced for the quality of estimate or filter. It is the level of damping random perturbations which is defined by the maximal value over all covariance matrices of the root-mean-square error normalized by the sum of variances of all random factors. The level of damping random perturbations was shown to be equal to the square of the spectral norm of the matrix relating the error of estimation and the random factors, and the optimal estimate minimizing this criterion was established. In the problem of filtration, it was shown how the filter parameters that are optimal in the level of damping random perturbations are expressed in terms of the linear matrix inequalities.
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Klimov, G.P., Teoriya veroyatnostei i matematicheskaya statistika (Probability Theory and Mathematical Statistics), Moscow: Mosk. Gos. Univ., 1983.
Fomin, V.N., Rekurrentnoe otsenivanie i adaptivnaya fil’tratsiya (Recurrent Estimation and Adaptive Filtration), Moscow: Nauka, 1984.
Pankov, A.R. and Semenikhin, K.V., Minimax Estimation by Probabilistic Criterion, Autom. Remote Control, 2007, vol. 68, no. 3, pp. 430–445.
Petersen, I.R. and McFarlane, D.C., Optimal Guaranteed Cost Filtering for Uncertain Discrete-Time Systems, Int. J. Robust Nonlin. Control, 1996, vol. 39, no. 6, pp. 267–280.
Zhu, X., Soh, Y.C., and Xie, L., Design and Analysis of Discrete-time Robust Kalman Filters, Automatica, 2002, vol. 38, pp. 1069–1077.
Dong, Z. and You, Z., Finite-horizon Robust Kalman Filtering for Uncertain Discrete Time-Varying Systems with Uncertain-Covariance White Noises, IEEE Signal Process. Lett., 2006, vol. 13, no. 8, pp. 493–496.
Mohamed, S.M.K. and Nahavandi, S., Robust Finite-Horizon Kalman Filtering for Uncertain Discrete-Time Systems, IEEE Trans. Automat. Control, 2012, vol. 57, pp. 1548–1552.
Poor, H.V. and Looze, D.P., Minimax State Estimation for Linear Stochastic Systems with Noise Uncertainty, IEEE Trans. Automat. Control, 1981, vol. 26, pp. 902–906.
Sayed, A.H., A Framework for State-Space Estimation with Uncertain Models, IEEE Trans. Automat. Control, 2001, vol. 46, pp. 998–1013.
Bitar, E., Baeyens, E., Packard, A., et al., Linear Minimax Estimation for Random Vectors with Parametric Uncertainty, in Proc. Am. Control Conf., 2010, pp. 590–592.
Kogan, M.M., LMI-based Minimax Estimation and Filtering under Unknown Covariances, Int. J. Control, 2014, vol. 87, no. 6, pp. 1216–1226.
Nagpal, K.M. and Khargonekar, P.P., Filtering and Smoothing in an H ∞ Setting, IEEE Trans. Automat. Control, 1991, vol. 36, no. 2, pp. 152–166.
Balandin, D.V. and Kogan, M.M., Generalized H ∞-optimal Control as a Trade-off between the H ∞-optimal and γ-optimal Controls, Autom. Remote Control, 2010, vol. 71, no. 6, pp. 993–1010.
Balandin, D.V. and Kogan, M.M., LMI-based H ∞-optimal Control with Transients, Int. J. Control, 2010, vol. 83, no. 8, pp. 1664–1673.
Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Design of the Control Laws on the Basis of Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.
Brammer, K. and Siffling, G., Kalman-Bucy Filter. Deterministische Beobachtung und stochastische Filterung, Munchen: Oldenbourg, 1975. Translated under the title Fil’tr Kalmana-B’yusi, Moscow: Nauka, 1982.
Albert, A., Regression and the Moor-Penrose Pseudoinverse, New York: Academic, 1972, Translated under the title Regressiya, psevdoinversiya i rekurrentnoe otsenivanie, Moscow: Nauka, 1977.
Wonham, W.M., Linear Multivariable Control: A Geometric Approach, New York: Springer, 1979. Translated under the title Linei’nye mnogomernye sistemy upravleniya: geometricheskii podkhod, Moscow: Nauka, 1980.
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Original Russian Text © M.M. Kogan, 2014, published in Avtomatika i Telemekhanika, 2014, No. 11, pp. 88–109.
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Kogan, M.M. Optimal estimation and filtration under unknown covariances of random factors. Autom Remote Control 75, 1964–1981 (2014). https://doi.org/10.1134/S000511791411006X
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DOI: https://doi.org/10.1134/S000511791411006X