A unifying framework for linear estimation: Generalized partitioned algorithms

• Authors: Demetrios G. Lainiotis
• Venue: Information Sciences: an International Journal
• Year: 1976

In this paper, generalized partitioned algorithms are presented that serve as the unifying framework for linear filtering and smoothing. The fundamental and all encompassing nature of the generalized partitioned algorithms (hereon denoted GPA) is clearly demonstrated by showing that the GPA contain as special cases important generalizations of past well-known linear estimation algorithms, as well as whole families of such algorithms, of which all previously obtained major filtering and smoothing algorithms are special cases. Specifically, generalized partitioned filtering and smoothing algorithms are given in terms of integral expressions, that are theoretically interesting, computationally attractive, as well as provide a unification of all previous approaches to linear filtering and smoothing, and clear delineation of their inter-relationships. In particular, the GPA for filtering are shown to contain as special cases families of filtering algorithms which constitute generalizations of the Kalman-Bucy, and Chandrasekhar algorithms as well as of the previously obtained partitioned algorithms of Lainiotis. These generalizations pertain to arbitrary initial conditions and time-varying models. Further, the GPA for smoothing are shown to contain two families of generalized backward and forward smoothing algorithms valid for arbitrary boundary conditions, of which all previous backward and forward algorithms are special cases. It is also shown that the GPA may also be given in terms of an imbedded generalized Chandrasekhar algorithm with the consequent computational advantages. Furthermore, the GPA are shown to serve as the basis of computationally effective, fast, and numerically robust algorithms for the numerical implementation of the estimation formulas. A particularly effective doubling algorithm is also given for calculating the steady-state filter. The partitioned numerical algorithms are given exactly in terms of a set of elemental solutions which are both simple as well as completely decoupled, and as such computable in either a parallel or serial processing mode. Moreover, the overall solution is given by a simple recursive operation on the elemental solutions.