Factorization Methods for Discrete Sequential Estimation

Chapter II is a review of the classical least squares problem. Recursive solutions of the Gauss normal equations are not only of historical interest, they are still used to solve problems in orbit determination, satellite navigation, missile tracking, economics, etc. Because of its prominent role in modern estimation we include a description of the Kalman filter data processing algorithm. It is ideally suited for real-tome updating of the least squares estimate and estimate error covariance and is the starting point for the development of covariance square root algorithms. Much of our SRIF development which is discussed in alter chapters was motivated by, and will be compared with, the Kalman filter., The chapter concludes with Potter's square toot mechanization of the Kalman algorithm. FORTRAN mechanizations of the Kalman and Potter algorithms are included as appendices.

Chapter III contains a review of various utilitarian properties of positive definite (PD) matrices that are useful for covariance analysis and are requisite for our development of square too estimation. FORTRAN mechanizations of several important PD algorithms are included as appendices.

In Chapter IV the Householder orthogonal transformation is introduced. Our SRIF development exploits the properties and algorithms that are described in this chapter.In Chapter V the orthogonal transformation material is used to develop a square root analogue of the normal equation recursions (cf. Chapter II). This result, due to Golub (1965), is the data processing portion of the SRIF. I this chapter we also introduce an alternative data processing algorithm based on a triangular square toot free factorization of the error covariance matrix. Our U-D Algorithm, (U and D being the upper triangular and diagonal factors, respectively) is a stable mechanization of a decomposition proposed by Agee and Turner (1972). The algorithm for UD^1/2 corresponds to the upper triangular covariance square root algorithm proposed by Carlson (1973). Computation count comparisons are mode of the Potter, Golub, and Carlson square root algorithms, the U-D factorization, and the Kalman filter. A FORTRAN mechanization of the U-D algorithm is included as a an appendix.

In Chapter VI the propagation portion of the SRIF, the Kalman filter, and the square root covariance filter are developed. These algorithms solve the prediction problem. This chapter also contains a discussion of the duality between covariance and information formulations of the filtering problem.

Chapter VII is an elaboration oft SRIF material covered in the previous two chapters. Computer implementation of the SIRF is expedited when the model is partitioned to distinguish the colored noise and bias parameters. Advantages accruing to such a partitioning are that computer storage requirements are significantly reduced and the special properties of the these variables result in additional analytic perceptions. In particular, the notions of sensitivity analysis, variable order filtering and consider covariance analysis carry over unchanged from those analyses that did not not involve process noise. This chapter also contains a colored noise processing algorithm that permits a unified treatment of colored and white noise. FORTRAN mechanizations of the various notions of this chapter are presented as an appendix. This computer mechanization used to design the filter portion of JPL's Orbit Determination Program, involves only a fraction of the storage and computation used in earlier SRIF computer implementations.

Chapters VIII and IX are devoted to the accuracy analysis and performance prediction aspects of filtering. Recursive algorithms are derived by analyzing the effects of incorrect a priori statistics and unmodeled parameters.

Chapter X focuses on smoothing, i.e., utilization of past and current measurement information to estimate the model's past history. Smoothing is important because many estimation applications do not involve (instantaneous) real-time decision making. Interesting and important features of our smoothing algorithm mechanization, which is an addendum to the SRIF, are that it requires only a modicum of additional computation and its structure is such that it preserves those features of the SRIF associated with sensitivity analysis, variable order estimation, and the consider covariance. In this chapter on smoothing we also review the covariance-related smoothing algorithms due to Rauch et al (1965) along with Bierman's (1973a) modifications of the Bryson-Frazier (1963) smoother.