Kalman filtering with constrained output injection

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 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 (Gelb 1974, pp. 104–109). 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. These reduced-complexity filters provide state estimates that are suboptimal relative to the classical Kalman filter (Bernstein and Hyland 1985, Hippe and Wurmthaler 1990, Haddad and Bernstein 1990, Hsieh 2003). Alternative variants of the classical Kalman filter have been developed for computationally demanding applications such as weather forecasting (Farrell and Ioannou 2001, Heemink et al. 2001, Ballabrera et al. 2001, Fieguth et al. 2003), where the classical Kalman filter gain and covariance are modified so as to reduce the computational requirements. The present paper is motivated by computationally demanding applications such as those discussed in Farrell and Ioannou (2001), Heemink et al. (2001), Ballabrera et al. (2001) and Fieguth et al. (2003). For such applications, a high-order simulation model is assumed to be available, but the derivation of a reduced-order filter in the sense of Bernstein and Hyland (1985), Hippe and Wurmthaler (1990), Haddad and Bernstein (1990), Hsieh (2003) is not feasible due to the high dimensionality of the analytic model. Instead, we use a full-order state estimator based directly on the simulation model. However, rather than implementing the classical Kalman filter, we derive an optimal spatially localized Kalman filter in which the structure of the filter gain is constrained 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, which is useful for variable-resolution discretizations of partial differential equations. Some of the results in this paper appeared in Barerro et al. (2005). The use of a spatially localized Kalman filter in place of the classical Kalman filter is also motivated by computational architecture constraints arising from a multiprocessor implementation of the Kalman filter (Lawrie et al. 1992) in which the Kalman filter operations can be confined to the subset of processors associated with the states whose estimates are desired. *Corresponding author. Email: [email protected]