State Estimation Using a Reduced-Order Kalman Filter

Brian F. Farrell Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

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Petros J. Ioannou Department of Physics, National and Capodistrian University of Athens, Athens, Greece

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Abstract

Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for a wide class of linear systems, and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time-dependent error covariance matrix, which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model, which suggests the use of reduced-order error models to obtain near-optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced-order approximation of the forecast error system. This reduced-order system is obtained by balanced truncation of the Hankel operator representation of the full error system and is used to construct a reduced-order Kalman filter for the purpose of state identification in a time-dependent quasigeostrophic storm track model. The accuracy of the state identification by the reduced-order Kalman filter is assessed by comparison to the true state, to the state estimate obtained by the full Kalman filter, and to the state estimate obtained by direct insertion.

Corresponding author address: Dr. Brian F. Farrell, Division of Engineering and Applied Sciences, Harvard University, Oxford St., Mail Area H0162, Cambridge, MA 02138. Email: farrell@deas.harvard.edu

Abstract

Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for a wide class of linear systems, and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time-dependent error covariance matrix, which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model, which suggests the use of reduced-order error models to obtain near-optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced-order approximation of the forecast error system. This reduced-order system is obtained by balanced truncation of the Hankel operator representation of the full error system and is used to construct a reduced-order Kalman filter for the purpose of state identification in a time-dependent quasigeostrophic storm track model. The accuracy of the state identification by the reduced-order Kalman filter is assessed by comparison to the true state, to the state estimate obtained by the full Kalman filter, and to the state estimate obtained by direct insertion.

Corresponding author address: Dr. Brian F. Farrell, Division of Engineering and Applied Sciences, Harvard University, Oxford St., Mail Area H0162, Cambridge, MA 02138. Email: farrell@deas.harvard.edu

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