Gerald J. Bierman

Brief paper: Measurement updating using the U-D factorization

In this paper we describe a fresh approach to the discrete linear filtering problem. Our method involves an upper triangular factorization of the filter error covariance matrix, i.e. P = UDUT. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is surprisingly simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Indeed, like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As in the case of the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.

Paper: Numerical comparison of kalman filter algorithms: Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.

Sequential square root filtering and smoothing of discrete linear systems

Square-root information estimation algorithms are immensely important estimation analysis tools that are not sufficiently well understood nor adequately exploited. In an endeavor to rectify this state of affairs an expository derivation of the square-root information filter/smoother is given. It is based on the recursive least-squares method and is easier to grasp, interpret and generalize than are the dynamic programming arguments previously used. Backward smoothing algorithms, both square-root and covariance recursions, are derived as direct and consequences of the method. A comparison of smoothing algorithms indicates that those presented in this paper are the most efficient. Partitioning the results to separate bias parameters provides further computational economies and reduction of storage requirements. The principal objective of this paper is to inspire greater utilization of square-root estimation algorithms. Arguments supporting this thesis are the new least-squares filter/smoother derivations, enhanced numerical accuracy, reduced computation, and lower storage requirements.

Fixed interval smoothing with discrete measurements

Abstract A recursive form of the Bryson-Frazier (1963) smoothing equations for a linear continuous dynamic system with linear discrete measurements is presented. The now algorithm exhibits a duality with the Kalman filtering equations. As a result of this duality it is possible to convert existing Kalman filtering programmes into smoothing programmes by redefining variables. Storage and computational requirements for the new algorithm are less than that for the Rauch-Tung-Streibel (1965) continuous smoother or the Fraser (1067) two-filter continuous smoother

Gram-Schmidt algorithms for covariance propagation

This paper addresses the time propagation of triangular covariance factors. Attention is focused on the square-root free factorization, P = UDUT, where U is unit upper triangular and D is diagonal. An efficient and reliable algorithm for U-D propagation is derived which employs Gram-Schmidt orthogonalization. Partitioning the state vector to distinguish bias and colored process noise parameters increases mapping efficiency. Cost comparisons of the U-D, Schmidt square-root covariance and conventional covariance propagation methods are made using weighted arithmetic operation counts. The U-D time update is shown to be less costly than the Schmidt method; and, except in unusual circumstances, it is within 20% of the cost of conventional propagation.

Measurement updating using the U-D factorization

In this paper we describe a fresh approach to the discrete linear filtering problem. Our method involves an upper triangular factorization of the filter error covariance matrix, i.e. P = UDUT. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is surprisingly simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Indeed, like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As in the case of the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.

Computational aspects of the matrix sign function solution to the ARE

Roberts matrix sign function solution to the ARE is defined so as to speed convergence and reduce storage requirements; our work extends ideas proposed by R. Byers, ref(8). Features of the sign function presented here are: (a) our formulation of the Roberts-Byers algorithm recurses on the symmetric transformed Hamiltonian, which reduces storage requirements, (b) the symmetric indefinite matrix inversion required by the algorithm is carried out using LINPACK (and our excellent numerical results reflect the wisdom of this choice); corrections to the computed (approximate) solution are obtained by applying the same algorithms to the translated problem (which improves upon the linear Lyapunov equation correction that has been used), and (d) simple (but somewhat ad hoc) convergence criteria are proposed to reduce computation. The algorithm described in this work has been tested on a variety of continuous time ARE test problems, and the results have been very satisfactory. Tests on numerically ill-conditioned problems produced results of comparable accuracy with those obtained by the Shur vector RICPACK method. Our sign function iterative ARE solution demonstrates numerical robustness, accurate results, rapid (super linear) convergence, algorithmic simplicity, and modest storage requirements. Our work shows that iterative ARE solutions offer a viable alternative to the Shur eigenvector approach that is a generally accepted reference.

UDUT Covariance Factorization for Kalman Filtering

There has been strong motivation to produce numerically stable formulations of the Kalman filter algorithms because it has long been known that the original discrete-time Kalman formulas are numerically unreliable. Numerical instability can be avoided by propagating certain factors of the estimate error covariance matrix rather than the covariance matrix itself. This paper documents filter algorithms that correspond to the covariance factorization P = UDU(T), where U is a unit upper triangular matrix and D is diagonal. Emphasis is on computational efficiency and numerical stability, since these properties are of key importance in real-time filter applications. The history of square-root and U-D covariance filters is reviewed. Simple examples are given to illustrate the numerical inadequacy of the Kalman covariance filter algorithms; these examples show how factorization techniques can give improved computational reliability.

Voyager orbit determination at Jupiter

This paper summarizes the Voyager 1 and Voyager 2 orbit determination activity extending from encounter minus 60 days to the Jupiter encounter, and includes quantitative results and conclusions derived from mission experience. The major topics covered include an identification and quantification of the major orbit determination error sources and a review of salient orbit determination results from encounter, with emphasis on the Jupiter approach phase orbit determination. Special attention is paid to the use of combined spacecraft-based optical observations and Earth-based radiometric observations to achieve accurate orbit determination during the Jupiter encounter approach phase.

Paper: A new computationally efficient fixed-interval, discrete-time smoother

The Rauch-Tung-Streibel smoother recursion is used to derive a new smoother algorithm based upon a decomposition of the linear model dynamical equation and maximizing use of rank-1 matrix modification. This new algorithm, it turns out, parallels Biermans forward recursive square-root information filter/ backward recursive U-D factorized covariance algorithm [5]. The new result features computational efficiency, reliance on numerically stable matrix modification algorithms, and reduced computer storage.