Spatially Localized Kalman Filtering for Data Assimilation Oscar Barrero*, Dennis S. Bernstein**, and Bart L. R. De Moor* Abstract— In data assimilation applications involving large scale systems, it is often of interest to estimate a subset of the system states. For example, for systems arising from discretized partial differential equations, the chosen subset of states can represent the desire to estimate state variables associated with a subregion of the spatial domain. The use of a spatially localized Kalman filter is motivated by computing constraints arising from a multi-processor implementation of the Kalman filter as well as a lack of global observability in a nonlinear system with an extended Kalman filter implemen- tation. In this paper we derive an extension of the classical output injection Kalman filter in which data is locally injected into a specified subset of the system states. I. INTRODUCTION The classical Kalman filter provides optimal least-squares estimates of all of the states of a linear time-varying system under process and measurement noise. In many applications, however, optimal estimates are desired for a specified subset of the system states, rather than all of the system states. For example, for systems arising from discretized partial differential equations, the chosen subset of states can represent the desire to estimate state variables associated with a subregion of the spatial domain. However, it is well known that the optimal state estimator for a subset of system states coincides with the classical Kalman filter. For applications involving high-order systems, it is often difficult to implement the classical Kalman filter, and thus it is of interest to consider computationally simpler filters that yield suboptimal estimates of a specified subset of states. One approach to this problem is to consider reduced- order Kalman filters. Such reduced-complexity controllers provide estimates of the desired states that are suboptimal relative to the classical Kalman filter [1–3, 6, 7]. Alternative variants of the classical Kalman filter have been developed for computationally demanding data assimilation applica- tions such as weather forecasting [8–10], where the classical Kalman filter gain and covariance are modified so as to reduce the computational requirements. The present paper is motivated by computationally de- manding applications such as those discussed in [8–10]. For such applications, a high-order simulation model is assumed to be available, and the derivation of a reduced- order filter in the sense of [1–3,6, 7] is not feasible due to the lack of a tractable analytic model. Instead, we consider This research was supported in part by the National Science Foundation under grant. *Department of Electrical Engineering, ESAT-SCD/SISTA, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium, obarrero@esat.kuleuven.ac.be, bart.demoor@esat.kuleuven.ac.be **Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140, USA, dsbaero@umich.edu the use of a full-order state estimator based directly on the simulation model. However, rather than implementing the classical output injection Kalman filter, we derive a suboptimal spatially localized Kalman filter in which the filter gain is constrained a priori to reflect the desire to estimate a specified subset of states. Our development is also more general than the classical treatment since the state dimension can be time varying. This extension is useful for variable-resolution discretizations of partial differential equations. The use of a spatially localized Kalman filter in place of the classical Kalman filter is motivated by the use of the ensemble Kalman filter for nonlinear systems. For sys- tems with sparse measurements, observability may not hold for the entire system. In this case, the spatially localized Kalman filter can be used to obtain state estimates for the observable portion of the system. II. SPATIALLY LOCALIZED KALMAN FILTER (SLKF) We begin by considering the discrete-time dynamical system x k = A k−1 x k−1 + B k−1 u k−1 + w k−1 , k ≥ 0, (1) with output y k = C k x k + v k , (2) where x k ∈ R n k , x k−1 ∈ R n k−1 , u k−1 ∈ R m k−1 , y k ∈ R l k , and A k−1 , B k−1 , C k are known real matrices of appropriate size. The input u k−1 and output y k are assumed to be measured, and w k−1 ∈ R n k−1 and v k ∈ R l k are zero-mean noise processes with known variances and correlation given by Q k−1 , R k , and S k , respectively. We assume that Q k−1 , R k , and S k are positive definite. Note that the dimension n k of the state x k can be time varying, and thus A k−1 ∈ R n k ×n k−1 is not necessarily square. The problem of estimating a subset of states of (1) from measurements of the output (2) is discussed in this section. A. Estimation Problem Consider the discrete-time dynamical system de- scribed by (1) and (2). For this system, we take a state estimator of the form ˆ x k|k = ˆ x k|k−1 + Γ k K k (y k − ˆ y k|k−1 ), k ≥ 0, (3) with output ˆ y k|k−1 = C k ˆ x k|k−1 . (4) where ˆ x k|k ∈ R n k is the estimation of x k using the mea- surements y i for 0 ≤ i ≤ k,ˆ y k|k−1 ∈ R l k , Γ k ∈ R n k ×p k , and K k ∈ R p k ×l k . The nontraditional feature of (3) is the presence 2005 American Control Conference June 8-10, 2005. Portland, OR, USA 0-7803-9098-9/05/$25.00 ©2005 AACC FrA02.6 3468