T. E. Bullock

Analysis of discrete-time Kalman filtering under incorrect noise covariances

Analysis tools are developed that can be effectively used to study the performance degradation of a filter when incorrect models of the state and measurement noise covariances are used. For a linear time-variant system with stationary noise processes, it is shown that under certain stability conditions on the system model, the one-step prediction error covariance matrix will converge to a steady-state solution even when the filter gain is not optimal. On the other hand, if the state transition matrix has an unreachable mode outside a unit circle, then the modeling errors in the noise covariances may cause the filter to diverge. Bounds on the asymptotic filter performance are computed when the range of errors in the noise covariance matrices are known. Using simple examples, insights into the behavior of a Kalman filter under nonideal conditions are provided. >

Technical communique: Invariant system description of the stochastic realization

In this article it is shown that the class of all realizations possessing the same power spectral density can be uniquely characterized by giving an invariant system description. A new transformation group is introduced and shown to leave the spectral density unchanged. The action of this group must be considered when attempting to specify a stochastic realization from spectral densities or equivalently covariance sequences.

Maneuver detection and tracking with a nonlinear target model

Two approaches to tracking a maneuverable target are presented. The tracking filter comprises two main parts: acceleration estimation and maneuver detection. The acceleration estimator is implemented first by assuming piecewise constant acceleration levels. Maneuver detection is performed to justify the validity of the assumption, i.e. if the acceleration has remained constant. If a maneuver is declared, the filter is reinitialized and a new level of constant acceleration is identified. The second approach is based on a nonlinear state model which gives better tracking for planar maneuvers and non constant accelerations.

Analysis of continuous-time Kalman filtering under incorrect noise covariances

Abstract The behavior of the continuous-time Kalman filter under incorrect noise covariances is analyzed. The filter performance is quantified by the actual state error covariance. Through this quantity, the characteristic of the filter is examined. Convergence and divergence analyses of the actual state error covariance are given. The significance of the results presented in the paper is that they help one to understand and be able to predict certain behavior of the Kalman filter when inexact values of noise covariances are used.

Observing a function of the state

This paper discusses a well known change of co-ordinates for multiple output systems from a different viewpoint, and a new, simplified, low order observer design. The discussion of the transformation given here exhibits more of the system structure than previous presentations. The observer design procedure does not require explicitly performing a change of basis on the system, and includes the construction of the linear transformation relating the system state to the asymptotic observer state.

Behavior of the discrete-time Kalman filter under incorrect noise covariances

In this paper, we study the behavior of the discrete-time Kalman filter under incorrect noise covariances. In particular, we are interested in the characteristic of the actual performance of the Kalman filter. The filter performance is quantified by the actual one-step predictor error covariance. Convergence and divergence analyses of the actual one-step predictor error covariance are given. The results developed in the paper provide useful insights in the behavior of the Kalman filter when the noise covariances used in designing the filter are inexact.

Mathematical analysis of bias in the extended Kalman filter

The state estimate measurement update process for the extended Kalman filter (EKF) as used in bearings-only estimation is investigated. Using a simplifying assumption it is shown mathematically that the gain and innovation sequences are correlated due to their joint dependence on the a priori cross-range estimation error. This correlation causes the range and range-rate estimates to be biased. Furthermore, it is shown that the modified gain EKF has the same state estimate update equation but that the gain is different due to a slightly different covariance update equation. Since the correlation between the gain and innovation sequences is not directly related to the covariance update, the claim that the modified gain EKF is unbiased is not substantiated. The difference in the covariance update equation does cause an alteration of the statistical properties of the gain sequence, but does not remove the correlation with the innovation sequence.<<ETX>>

Reachable Set Control For Preferred Axis Homing Missiles

This paper presents the results of a linear control technique developed for fixed-final-time constrained systems and applied to preferred axis homing missiles. This technique accommodated the nonlinearities, constraints, and the modeling errors of a suboptimal linear design to maintain the target within the missile reachable set. This feature minimized miss distances, and prevented control saturation during the terminal portion of the trajectory.

Hybrid computer integration of partial differential equations by use of an assumed sum seperation of variables

The majority of numerical solution methods for partial differential equations by either analog or digital methods involve some form of finite differences technique, integral transforms, or Monto-Carlo methods. On the other hand, the most common classical analytical approach is based on some form of separation of variables and series expansions. The motivation for the research presented in this paper was to investigate the possibility of using the classical separation of variables approach as a basis for an efficient computational algorithm. The method studied was developed with a hybrid computer implemention in mind due to the ease in on-line operation in engineering design applications although it could be used for digital computation also.

A federated extended Kalman filter with application to passive target tracking

The problem of determining the track of a maneuvering target from passive measurements of the sensor to target line-of-sight is addressed. The standard approach to such an estimation problem, the extended Kalman filter (EKF), suffers from a range and range-rate bias that limits its usefulness. The cause of this bias has been identified as a correlation between the gain and innovations sequences that is inherent in the EKF. These previous results are presented and then a filter design is developed that reduces this unwanted effect. Simulation results reveal a performance improvement using the new filter design.<<ETX>>