Multi-Partitioning Linear Estimation Algorithms: Discrete Time Systems

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The partitioned estimation aloorithms of Lainiotis for the linear discrete time state estimation problems have been generalized in two important ways. First, the inital condition of the estimation problem can be partitioned into the sum of an arbitrary number of jointly Gaussian random variables; and second, these jointly Gaussian random variables may be statistically dependent. The form of the resulting algorithm consists of an imbedded Kalman filter with partial initial conditions and one correction term for each other partition or subdivision of the initial state vector.This approach, called multi-partitioning, can be used to provide added insight into the estimation problem. One significant application is in the parameter identification problem where identification algorithms can be formulated in which the inversion of the information matrix of the parameters is replaced by simple division by scalars. A second use of multi-partitioning is to show the specific effects on the filtered state estimate of off-diagonal terms in the initial-state covariance matrix.An extremely important offshoot of the multi-partitioning results is a new formula for the inversion of a symmetric matrix. The nature of the solution shows promise for the inversion of numerically ill conditioned matrices. This result is applicable not only to estimation problems but to matrix theory in general.