We propose a general reduced-order filtering strategy adapted to Unscented Kalman Filtering for any choice of sampling points distribution. This provides tractable filtering algorithms which can be used with large-dimensional systems when the uncertainty space is of reduced size, and these algorithms only invoke the original dynamical and observation operators, namely, they do not require tangent operator computations, which of course is of considerable benefit when nonlinear operators are considered. The algorithms are derived in discrete time as in the classical UKF formalism - well-adapted to time discretized dynamical equations - and then extended into consistent continuous-time versions. This reduced-order filtering approach can be used in particular for the estimation of parameters in large dynamical systems arising from the discretization of partial differential equations, when state estimation can be handled by an adequate Luenberger observer inspired from feedback control. In this case, we give an analysis of the joint state-parameter estimation procedure based on linearized error, and we illustrate the effectiveness of the approach using a test problem inspired from cardiac biomechanics.

Classification: 93E11, 93B30, 35R30, 74H15

Keywords: filtering, data assimilation, state and parameter estimation, identification in PDEs

@article{COCV_2011__17_2_380_0, author = {Moireau, Philippe and Chapelle, Dominique}, title = {Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {17}, number = {2}, year = {2011}, pages = {380-405}, doi = {10.1051/cocv/2010006}, zbl = {1243.93114}, mrnumber = {2801324}, language = {en}, url = {http://www.numdam.org/item/COCV_2011__17_2_380_0} }

Moireau, Philippe; Chapelle, Dominique. Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 380-405. doi : 10.1051/cocv/2010006. http://www.numdam.org/item/COCV_2011__17_2_380_0/

[1] AHA/ACC/SNM, Standardization of cardiac tomographic imaging. Circulation 86 (1992) 338-339.

[2] Tagged magnetic resonance imaging of the heart: a survey. Med. Image Anal. 9 (2005) 376.

, and ,[3] Finite Element Procedures. Prentice-Hall, USA (1996). | Zbl 0994.74001

,[4] Data assimilation for geophysical fluids, in Handbook of Numerical Analysis: Computational Methods for the Atmosphere and the Oceans, R. Temam and J. Tribbia Eds., Elsevier (2008). | MR 2454278

, and ,[5] Mapping tropical Pacific sea level: Data assimilation via a reduced state space Kalman filter. J. Geophys. Res. 101 (1996) 22599-22618.

, , , , and ,[6] Identification of Robin coefficients by the means of boundary measurements. Inv. Prob. 15 (1999) 1425-1438. | MR 1733209 | Zbl 0943.35100

and ,[7] Robust filtering for joint state-parameter estimation in distributed mechanical systems. DCDS-A 23 (2009) 65-84. | MR 2449069 | Zbl 1154.35462

, and ,[8] Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR 2487899 | Zbl 1163.74019

and ,[9] A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific. J. Mar. Syst. 36 (2002) 101-127.

, and ,[10] Reduced Sigma Point Filters for the Propagation of Means and Covariances through Nonlinear Transformations, in Proc. of IEEE Am. Contr. Conf., Anchorage AK, USA, 8-10 May (2002) 887-892.

and ,[11] The Scaled Unscented Transformation, in Proc. of IEEE Am. Contr. Conf., Anchorage AK, USA, 8-10 May (2002) 4555-4559.

and ,[12] A new approach for filtering nonlinear systems, in American Control Conference (1995) 1628-1632.

, and ,[13] A new method for the nonlinear transformation of means and covariances in filter and estimators. IEEE Trans. Automat. Contr. 45 (2000) 447-482. | MR 1762859 | Zbl 0973.93053

, and ,[14] Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis 3, P.G. Ciarlet and J.-L. Lions Eds., Elsevier (1994). | MR 1307410 | Zbl 0875.73234

,[15] Comments on “A new method for the nonlinear transformation of means and covariances in filters and estimators” [and authors' reply]. IEEE Trans. Automat. Contr. 47 (2002) 1406- 1409. | MR 1917460 | Zbl 0973.93053

, and ,[16] An introduction to observers. IEEE Trans. Automat. Contr. 16 (1971) 596-602.

,[17] Joint state and parameter estimation for distributed mechanical systems. Comput. Meth. Appl. Mech. Eng. 197 (2008) 659-677. | MR 2397009 | Zbl 1169.74439

, and ,[18] Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging. Inv. Prob. 25 (2009) 035010. | MR 2480180 | Zbl 1169.35393

, and ,[19] Filtres de Kalman singuliers évolutifs pour l'assimilation de données en océanographie. C. R. Acad. Sci. - Ser. IIA 326 (1998) 255-260.

, and ,[20] A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Marine Systems 16 (1998) 323-341.

, and ,[21] On unscented Kalman filtering for state estimation of continuous-time nonlinear systems. IEEE Trans. Automat. Contr. 52 (2007) 1631-1641. | MR 2352439

,[22] Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches. Wiley-Interscience (2006).

,[23] Application of the unscented Kalman filter for real-time nonlinear structural system identification. Struct. Contr. Health. Monit. 14 (2006) 971-990.

and ,[24] Adaptive observer with exponential forgetting factor for linear time varying systems, in Proceedings of the 40th IEEE Conference on Decision and Control 4 (2001) 3886-3891.

and ,