Milton B. Adams

Linear estimation of boundary value stochastic processes-- Part I: The role and construction of complementary models

This paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by Weinert and Desai in their important paper [1]. Specifically, the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions. A major contribution of this paper is an application of Greens identity in deriving a differential operator representation of the estimator. To clarify the development and to illustrate the range of applications of our approach, two brief examples are provided: one is a 1-D discrete two-point boundary value process and the other is a 2-D process governed by Poissons equation on the unit disk.

Solution and linear estimation of 2-D nearest-neighbor models

The solution and linear estimation of 2-D nearest-neighbor models (NNMs) are considered. The class of problems that can be described by NNMs is quite large, as models of this type arise whenever partial differential equations are discretized with finite-difference methods. A general solution technique that relies on converting the system to an equivalent 1-D two-point boundary-value descriptor system (TPBVDS) of large dimension, for which a recursive and stable solution technique is developed, is proposed. Under slightly restrictive assumptions, an even faster procedure can be obtained by using the fast Fourier transform (FFT) with respect to one of the space dimensions to convert the 1-D TPBVDS into a set of decoupled TPBVDS of low order, which can be solved in parallel. The smoothing problem for 2-D random fields described by stochastic NNMs is also examined. The smoother is expressed as a Hamiltonian system of twice the dimension of the original system and is also in NNM form. NNM solution techniques are therefore directly applicable to this solution. The results are illustrated by two examples corresponding to the discretized Poisson and heat equations, respectively. >

Linear estimation of boundary value stochastic processes--Part II: 1-D smoothing problems

This paper addresses the fixed-interval smoothing problem for linear two-point boundary value stochastic processes of the type introduced by Krener [5]. As these models are not Markovian, Kalman filtering and associated smoothing algorithms are not applicable. The smoothing problem for this class of noncausal processes is solved here by an application of the estimator solution which is developed in Part I of this paper [3] via the method of complementary models. For an n th-order model, this approach yields the smoother as a 2 n th-order two-point boundary value problem. It is shown that this smoother can be realized in a stable two-filter form which is remarkably similar to two-filter smoothers for causal processes. In addition, expressions for the smoothing error and smoothing error covariance are developed. These equations are employed to perform a covariance analysis of estimating the temperature and heat flow in a cooling fin.

Estimation for boundary-value descriptor systems

In this paper we consider models for noncausal processes consisting of discrete-time descriptor dynamics and boundary conditions on the values of the process at the two ends of the interval on which the process is defined. We discuss the general solution and well-posedness of systems of this type and then apply the method of complementary processes to obtain a specification of the optimal smoother in terms of a boundary-value descriptor Hamiltonian system. We then study the implementation of the optimal smoother. Motivated by the Hamiltonian diagonalization results for nondescriptor systems, we show how the descriptor Hamiltonian dynamics can be transformed to two lower-order systems by the use of transformation matrices involving the solution of two generalized Riccati equations. We present several examples illustrating our results and the nature of the smoothing solution and also present equations for covariance analysis of boundary-value descriptor processes including the smoothing error.

Attitude and Oribit Estimation Using Stars and Landmarks

An extended Kalman filter is used to process line-of-sight measurements to stars and known landmarks providing a statistical indication of performance in estimating spacecraft attitude, orbital ephemeris, and the bias drift of a set of three strapdown gyros. The landmark measurements were assumed to have been taken from the imagery of an Earth-observing multispectral scanner. It is shown that filtering of these noisy measurements results in highly accurate estimates of the above parameters. Results are given showing the sensitivity of performance to various system parameters such as star tracker accuracy, errors in the knowledge of landmark position, and number of stars and landmarks processed.

Determination of Failure Thresholds in Hybrid Navigation

A systematic approach for the determination of failure thresholds for hybrid navigation systems is described. Cost functions which reflect the importance assigned to the consequences of false and missed alarms are minimized. The false alarm probability is obtained as a function of the threshold magnitude by observing the statistical behavior of the instrument outputs in the normal operating mode. The missed alarm probability is obtained by determining the sensitivity of navigation error performance to instrument error sources. Two cost functions are considered. To illustrate this method, failure detection and identification (FDI) thresholds are determined for the Space Shuttle Approach and Landing Test flight.

Linear smoothing for descriptor systems

The linear smoother for an n-th order two-point boundary value descriptor system (TPBVDS) [1] is formulated by using the general linear estimator solution developed by the method of complementary models in [2]. The smoother is shown to take the form of a 2nth order TPBVDS. By employing the solutions to generalized Riccati equations [3] it is shown that the smoother dynamics can be decomposed into two nth order descriptor form equations. The implementation of the smoother solution is also addressed.

Real-time mission and trajectory planning

An automated mission and trajectory planner for piloted as well as autonomous vehicle applications is described. The planning system is composed of a trajectory planner and a mission planner. The trajectory planner generates multiple, near-optimal trajectories between mission objectives and estimates the costs (in terms of time, energy and lethality) to fly those trajectories. The mission planner uses this information to select and order the subset of mission objectives that is consistent with any constraints that have been imposed on the planning process. By viewing the time available to plan as an explicit constraint on the planning process, the planner is capable of adapting its planning strategy to the requirements of a broad spectrum of real-time applications. The automated mission and trajectory planner has been implemented and evaluated in a serial computational environment. Results are presented to illustrate the capabilities of the planner in the context of a hypothetical vehicle and mission environment.

Linear estimation of boundary value processes

In this paper we discuss the problem of estimating boundary value processes in one or several dimensions. The estimator dynamics are described, and by using operator transformations for these dynamics, several implementations are obtained which either diagonalize or triangularize the linear least-squares estimator. These implementations enable us to compute the estimate of the process by using two-filter type of smoothing formulas, or more general smoothing formulas similar to those used for solving the smoothing problem for 1-D causal processes.

MODELING ALTERNATIVE AIR TRAFFIC FLOW MANAGEMENT CONCEPTS

Abstract This paper describes a range of alternative concepts for the policies and procedures under which the ATFM system operates and a set of integrated models needed to examine their costs and benefits. The concepts represent a multi-stage evolution from the current system to increasingly decentralized approaches to decision-making.