×

Ellipsoidal parameter or state estimation under model uncertainty. (English) Zbl 1056.93063

Summary: Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.

MSC:

93E10 Estimation and detection in stochastic control theory
93E12 Identification in stochastic control theory
93B40 Computational methods in systems theory (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ben-Tal, A.; El Ghaoui, L.; Nemirovskii, A.S., Robust semidefinite programming, (), 139-162
[2] Bertsekas, D.P.; Rhodes, I.B., Recursive state estimation for a set-membership description of uncertainty, IEEE transactions on automatic control, 16, 117-128, (1971)
[3] Boyd, S.; El Ghaoui, L.; Ferron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia
[4] Cerone, V., Feasible parameter set for linear models with bounded errors in all variables, Automatica, 29, 1551-1555, (1993) · Zbl 0800.93237
[5] Chernousko, F.L., Optimal guaranteed estimates of indeterminacies with the aid of ellipsoids. I-III, Engineering cybernetics, 18, 3-5, (1981)
[6] Chernousko, F.L., State estimation for dynamic systems, (1994), CRC Press Boca Raton · Zbl 0830.93032
[7] Chernousko, F.L., Ellipsoidal approximation of attainability sets for linear system with uncertain matrix (in Russian), Prikladnaya matematika i mekhanika, 80, 940-950, (1996)
[8] Chernousko, F.L.; Rokityanskii, D.Ya., Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations, Journal of optimization theory and applications, 104, 1-19, (2000) · Zbl 0968.93011
[9] Clement, T.; Gentil, S., Recursive membership set estimation for output-errors models, Mathematics and computers in simulation, 32, 505-513, (1990)
[10] Durieu, C.; Walter, E.; Polyak, B.T., Multi-input multi-output ellipsoidal state bounding, Journal of optimization theory and applications, 111, 273-303, (2001) · Zbl 1080.93656
[11] El Ghaoui, L.; Calafiore, G., Worst-case simulation of uncertain systems, (), 134-146 · Zbl 0948.93506
[12] El Ghaoui, L.; Lebret, H., Robust solutions to least-squares problems with uncertain data, SIAM journal of matrix analysis and applications, 18, 1035-1064, (1997) · Zbl 0891.65039
[13] Fogel, E.; Huang, Y.F., On the value of information in system identification—bounded noise case, Automatica, 18, 229-238, (1982) · Zbl 0433.93062
[14] Golub, G.H.; Van Loan, C.F., An analysis of the total least squares problems, SIAM journal on numerical analysis, 17, 883-893, (1980) · Zbl 0468.65011
[15] Kurzhanskii, A. B. (1977). Control and observation under uncertainty (in Russian). Moscow: Nauka. · Zbl 0461.93001
[16] Kurzhanskii, A.B.; Valyi, I., Ellipsoidal calculus for estimation and control, (1997), Birkhauser Boston
[17] Milanese, M., Norton, J. P., Piet-Lahanier, H., & Walter, E. (Eds.) (1996). Bounding approaches to system identification. New York: Plenum. · Zbl 0845.00024
[18] Norton, J. P. (Ed.) (1994, 1995). Special issues on bounded-error estimation. I. II. International Journal of Adaptive Control and Signal Processing, 8(1), 9(2).
[19] Norton, J.P., Modal robust state estimator with deterministic specification of uncertainty, (), 62-71 · Zbl 0949.93506
[20] Polyak, B.T., Convexity of quadratic transformations and its use in control and optimization, Journal of optimization theory and applications, 99, 553-583, (1998) · Zbl 0961.90074
[21] Rokityanskii, D.Ya., Optimal ellipsoidal estimates of attainability sets for linear systems with uncertain matrix (in Russian), Izvestiya RAN. theor. i syst. upr., 4, 17-20, (1997)
[22] Schweppe, F.C., Recursive state estimation: unknown but bounded errors and system inputs, IEEE transactions on automatic control, 13, 22-28, (1968)
[23] Schweppe, F.C., Uncertain dynamic systems, (1973), Prentice-Hall Englewood Cliffs, NJ
[24] Walter, E. (Ed.) (1990). Special issue on parameter identification with error bound. Mathematics and Computers in Simulations, 32(5-6).
[25] Walter, E.; Pronzato, L., Identification of parametric models from experimental data, (1997), Springer London
[26] Witsenhausen, H.S., Sets of possible states of linear systems given perturbed observations, IEEE transactions on automatic control, 13, 5, 556-558, (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.