A unifying approach to linear estimation via the partitioned algorithms, I: Continuous models

Abstract

In this paper, the fundamental nature of the "partitioned" algorithms is demonstrated by showing that the "partitioned" algorithms serve as the basis of a unifying approach to linear filtering and smoothing. Specifically, generalized "partitioned" filtering and smoothing algorithms are given in terms of forward and backward-time differentiations that are theoretically interesting, possibly computationally attractive, as well as provide a unification of the previous major approaches to filtering and smoothing and clear delineation of their inter-relationships. In particular, the generalized "partitioned" filtering algorithms are shown to contain as special cases both the Kalman-Bucy filter as well as the Chandrasekhar algorithms. Furthermore, the generalized "partitioned" algorithms lead to important generalizations of the Chandrasekhar algorithms [5-7, 18- 19], as well as of the previous "partitioned" algorithms of the author [15-19]. These generalizations pertain to arbitrary initial conditions and time-varying models. It is also shown [20-22] that the generalized "partitioned" algorithm may also be given in terms of an imbedded generalized Chandrasekhar algorithm with the consequent possible computational advantages. Similarly, the generalized "partitioned" smoothing algorithm is shown to contain as special cases the major algorithms for smoothing such as those of Mayne and Fraser [9-10], Kailath and Frost [12], and Meditch [11], as well as the related ones of Zachrisson [13], and Biswas and Mahalanobis [14]. Finally, backwards smoothing algorithms for arbitrary boundary conditions are also obtained as a special case of the "partitioned" smoothing algorithms.