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Robust estimation and filtering in uncertain linear systems under unknown covariations

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Abstract

In the problems of estimation and filtering under uncertainty in the regressors, parameters, and covariances of random noise and perturbations, the best possible upper boundary of the proportionality coefficient between the root-mean-square error of estimate or filter and the sum of variances of all random factors was determined. This boundary which was named the level of suppression of random perturbations is characterized in terms of the linear matrix inequalities. The minimax estimate and minimax filter optimizing this index were established. The optimal robust estimate and filter were obtained using additional information about the membership of the covariance matrix in the given convex polyhedron.

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References

  1. 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.

    Article  MATH  MathSciNet  Google Scholar 

  2. Sayed, A.H., A Framework for State-space Estimation with Uncertain Models, IEEE Trans. Automat. Control, 2001, vol. 26, pp. 998–1013.

    Article  MathSciNet  Google Scholar 

  3. Calafiore, G. and El Ghaoui, L., Minimum Variance Estimation with Uncertain Statistical Model, in Proc. IEEE CDC, 2001, pp. 3497–3499.

    Google Scholar 

  4. Zhu, X., Soh, Y.C., and Xie, L., Design and Analysis of Discrete-time Robust Kalman Filters, Automatica, 2002, vol. 38, pp. 1069–1077.

    Article  MATH  MathSciNet  Google Scholar 

  5. 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.

    Article  Google Scholar 

  6. Bitar, E., Baeyens, E., Packard, A., et al., Minimax Estimation for Random Vectors with Parametric Uncertainty, in Proc. Linear Am. Control Conf., 2010, pp. 590–592.

    Google Scholar 

  7. Pankov, A.R., Platonov, E.N., and Semenikhin, K.V., Robust Filtering of Process in the Stationary Difference Stochastic System, Autom. Remote Control, 2011, vol. 72, no. 2, pp. 377–392.

    Article  MATH  MathSciNet  Google Scholar 

  8. Mohamed, S.M.K. and Nahavandi, S., Robust Finite-horizon Kalman Filtering for Uncertain Discretetime Systems, IEEE Trans. Automat. Control, 2012, vol. 57, pp. 1548–1552.

    Article  MathSciNet  Google Scholar 

  9. Fu, M., de Souza, C.E., and Xie, L., H ∞ Estimation for Uncertain Systems, Int. J. Robust Nonlin. Control, 1992, vol. 2, pp. 87–105.

    Article  MATH  Google Scholar 

  10. Xie, L., Soh, Y.C., and de Souza, C.E., Robust Kalman Filtering for Uncertain Discrete-time Systems, IEEE Trans. Automat. Control, 1994, vol. 39, no. 6, pp. 1310–1314.

    Article  MATH  MathSciNet  Google Scholar 

  11. Khargonekar, P.P., Rotea, M.A., and Baeyens, E., Mixed H 2/H ∞ Filtering, Int. J. Robust Nonlin. Control, 1996, vol. 6, pp. 313–330.

    Article  MATH  MathSciNet  Google Scholar 

  12. Geromel, J.C., Oliveira, M.C., and Bernussou, J., Robust Filtering of Discrete-time Linear Systems with Parameter Dependent Lyapunov Functions, in Proc. IEEE CDC, 1999, pp. 570–575.

    Google Scholar 

  13. Garsia G., Tarbouriech S., Peres P.L.D., Robust Kalman Filtering for Uncertain Discrete-time Linear Systems, Int. J. Robust Nonlin. Control, 2003, vol. 13, pp. 1225–1238.

    Article  Google Scholar 

  14. Duan, Z., Zhang, J., Zhang, C., et al., Robust H 2 and H ∞ Filtering for Uncertain Linear Systems, Automatica, 2006, vol. 42, pp. 1919–1926.

    Article  MATH  MathSciNet  Google Scholar 

  15. Petersen, I.R. and McFarlane, D.C., Optimal Guaranteed Cost Filtering for Uncertain Discrete-time Linear Systems, Int. J. Robust Nonlin. Control, 1996, vol. 6, pp. 267–280.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kogan, M.M., LMI-based Minimax Estimation and Filtering under Unknown Covariances, Int. J. Control, 2014, vol. 87, no. 6, pp. 1216–1226.

    Article  MATH  Google Scholar 

  17. Kogan, M.M., Optimal Estimation and Filtration under Unknown Covariances of Random Factors, Autom. Remote Control, 2014, vol. 75, no. 11, pp. 1964–1981.

    Article  MathSciNet  Google Scholar 

  18. Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994

    Book  MATH  Google Scholar 

  19. Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Design of Control Laws on the Basis of Linear Matrix Inequalities), Moscow: Fizmatlit, 2007

    Google Scholar 

  20. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh (tekhnika lineinykh matrichnykh neravenstv) (Control of Linear Systems under External Perturbations (Procedure of Linear Matrix Inequalities)), Moscow: LELAND, 2014

    Google Scholar 

  21. Petersen, I.R., A Stabilization Algorithm for a Class of Uncertain Systems, Syst. Control Lett., 1987, vol. 8, pp. 351–357.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Nizhny Novgorod State University of Architecture and Civil Engineering, Nizhny Novgorod, Russia

    M. M. Kogan

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Correspondence to M. M. Kogan.

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Original Russian Text © M.M. Kogan, 2015, published in Avtomatika i Telemekhanika, 2015, No. 10, pp. 50–66.

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Kogan, M.M. Robust estimation and filtering in uncertain linear systems under unknown covariations. Autom Remote Control 76, 1751–1764 (2015). https://doi.org/10.1134/S0005117915100033

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