# Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems

Philippe Moireau; Dominique Chapelle

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 2, page 380-405
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topMoireau, Philippe, and Chapelle, Dominique. "Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 380-405. <http://eudml.org/doc/272963>.

@article{Moireau2011,

abstract = {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.},

author = {Moireau, Philippe, Chapelle, Dominique},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {filtering; data assimilation; state and parameter estimation; identification in PDEs},

language = {eng},

number = {2},

pages = {380-405},

publisher = {EDP-Sciences},

title = {Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems},

url = {http://eudml.org/doc/272963},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Moireau, Philippe

AU - Chapelle, Dominique

TI - Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 2

SP - 380

EP - 405

AB - 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.

LA - eng

KW - filtering; data assimilation; state and parameter estimation; identification in PDEs

UR - http://eudml.org/doc/272963

ER -

## References

top- [1] AHA/ACC/SNM, Standardization of cardiac tomographic imaging. Circulation 86 (1992) 338–339.
- [2] L. Axel, A. Montillo and D. Kim, Tagged magnetic resonance imaging of the heart: a survey. Med. Image Anal. 9 (2005) 376.
- [3] K.J. Bathe, Finite Element Procedures. Prentice-Hall, USA (1996). Zbl0994.74001
- [4] J. Blum, F.X. Le Dimet and I.M. Navon, 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). MR2454278
- [5] M.A. Cane, A. Kaplan, R.N. Miller, B. Tang, E.C. Hackert and A.J. Busalacchi, Mapping tropical Pacific sea level: Data assimilation via a reduced state space Kalman filter. J. Geophys. Res.101 (1996) 22599–22618.
- [6] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements. Inv. Prob.15 (1999) 1425–1438. Zbl0943.35100MR1733209
- [7] D. Chapelle, P. Moireau and P. Le Tallec, Robust filtering for joint state-parameter estimation in distributed mechanical systems. DCDS–A23 (2009) 65–84. Zbl1154.35462MR2449069
- [8] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl.91 (2009) 20–48. Zbl1163.74019MR2487899
- [9] I. Hoteit, D.-T. Pham and J. Blum, A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific. J. Mar. Syst.36 (2002) 101–127.
- [10] S.J. Julier and J.K. Uhlmann, 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.
- [11] S.J. Julier and J.K. Uhlmann, The Scaled Unscented Transformation, in Proc. of IEEE Am. Contr. Conf., Anchorage AK, USA, 8–10 May (2002) 4555–4559.
- [12] S. Julier, J. Uhlmann and H. Durrant-Whyte, A new approach for filtering nonlinear systems, in American Control Conference (1995) 1628–1632.
- [13] S. Julier, J. Uhlmann and H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filter and estimators. IEEE Trans. Automat. Contr.45 (2000) 447–482. Zbl0973.93053MR1762859
- [14] P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis 3, P.G. Ciarlet and J.-L. Lions Eds., Elsevier (1994). Zbl0875.73234MR1307410
- [15] T. Lefebvre, H. Bruyninckx and J. De Schuller, 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. Zbl0973.93053MR1917460
- [16] D.G. Luenberger, An introduction to observers. IEEE Trans. Automat. Contr.16 (1971) 596–602.
- [17] P. Moireau, D. Chapelle and P. Le Tallec, Joint state and parameter estimation for distributed mechanical systems. Comput. Meth. Appl. Mech. Eng.197 (2008) 659–677. Zbl1169.74439MR2397009
- [18] P. Moireau, D. Chapelle and P. Le Tallec, Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging. Inv. Prob. 25 (2009) 035010. Zbl1169.35393MR2480180
- [19] D.-T. Pham, J. Verron and L. Gourdeau, Filtres de Kalman singuliers évolutifs pour l'assimilation de données en océanographie. C. R. Acad. Sci. – Ser. IIA326 (1998) 255–260.
- [20] D.T. Pham, J. Verron and M.C. Roubeaud, A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Marine Systems16 (1998) 323–341.
- [21] S. Sarkka, On unscented Kalman filtering for state estimation of continuous-time nonlinear systems. IEEE Trans. Automat. Contr.52 (2007) 1631–1641. MR2352439
- [22] D. Simon, Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches. Wiley-Interscience (2006).
- [23] M. Wu and A.W. Smyth, Application of the unscented Kalman filter for real-time nonlinear structural system identification. Struct. Contr. Health. Monit.14 (2006) 971–990.
- [24] Q. Zhang and A. Clavel, 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.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.