Showing papers on "Kalman filter published in 1972"

Approaches to adaptive filtering

Abstract: The different methods of adaptive filtering are divided into four categories: Bayesian, maximum likelihood (ML), correlation, and covariance matching. The relationship between the methods and the difficulties associated with each method are described. New algorithms for the direct estimation of the optimal gain of a Kalman filter are given.

Two-dimensional Bayesian estimate of images

Abstract: A dynamic model for pictorial data that can be represented by a random field of an exponential autocorrelation function is developed. A partial difference equation describes the dynamic model and is used to realize a two-dimensional recursive filter that gives a Bayesian-estimate of the pictorial data from a noisy observation of the data. It is assumed that the noise is additive, white, and uncorrelated with the signal. Practical application of the estimation technique is illustrated by applying the results to enhance several pictures. A comparison of this technique and its one-dimensional counterpart (Kalman filter) is made, and generalization of the estimation technique to other autoregressive sources is considered.

Application of Kalman Filtering to the Surveillance and Control of Traffic Systems

Abstract: The methodology of the discrete-time, extended Kalman filter is applied for the estimation of densities and the control of critical traffic links. The methodology is tested using traffic data obtained at the Lincoln tunnel of New York City. Two algorithms are tested, one involving density estimation alone and one combining density estimation with a formalism for the determination of optimal control. The results indicate that the first algorithm gives very good density estimates. The second algorithm yields a less accurate density estimate, but has the advantage over the first that it is amenable to an analytical optimization investigation.

Bayesian recursive image estimation.

Abstract: A procedure for recursively estimating images that are characterized statistically by the mean and correlation functions associated with the random process representing the brightness level is proposed for the case where the images are corrupted by additive noise. First, a dynamic model is developed with a response characteristic which matches that of the scanner output (the input of the estimator is the output of a horizontal line scanner) in a statistical sense. Such models have the form of an ordinary differential or difference equation with white noise input. An insignificant approximation is introduced by using a constant-coefficient model. The appropriate model is a vector valued difference equation with the solution representing a vector Markov process. The next step is to obtain the minimum mean square estimate of the image by using a Kalman filter. Since the image estimation is an interpolation problem, two successive runs over the observation are performed in opposite directions and the resultant estimates are averaged. Examples are included for illustration.

Integrated Navigation Systems and Kalman Filtering: A Perspective

Abstract: Kalman filtering has been used in a wide variety of aided inertial navigation systems in recent years Yet it appears that Kalman filtering and its overall role in the integrated system is still not well understood by many in the navigation community This paper is largely tutorial and is directed toward pinpointing the precise role that the filter plays in the integrated system The presentation is made from a systems viewpoint with the details of Kalman filtering completely surpressed It is first noted that in the current generation of aided inertial systems the filter operates only on the system errors and not on total dynamical quantities such as position and velocity The inertial system is then corrected in accordance with the filter's best estimates of the system errors It is shown that this mode of operation fits within the framework of complementary filtering which has been used in a number of instrumentation applications This perspective is particularly useful in helping one understand the system limitations in this mode of operation Next, it is pointed out that the more general problem of estimating total position and velocity is actually one of nonlinear estimation Within this context, then, the current scheme of system integration can be seen as a special form of nonlinear filtering known as an extended Kalman filter When viewed this way, one thinks of the inertial system as providing the estimated trajectory, and the aiding sources are the noisy measurements that provide corrections to the trajectory This viewpoint gives some additional insight into the filter limitations, because nonlinear estimation theory can be brought to bear on the problem In summary, the current state of the art is on a plateau in terms of system organization, and has been for the past decade No genuinely new information-processing concepts have been introduced into the navigation business since the early Sixties The paper concludes on a speculative note with regard to possible advances in system organization

Bayesian recursive image estimation.

Abstract: The enhancement of images that are characterized only by statistical data where the picture contains additive noise is considered. The random process representing the output of the scanner is characterized by the output of a dynamic system with white noise input. The dynamic system describes a first-order vector-Markov process. The procedure of Kalman filtering is then utilized to recursively determine the minimum mean-square error estimate of the image. The result is then extended to obtain the smoothing of the data.

Bayesian approach to distributed-parameter filtering and smoothing†

Abstract: The filtering and smoothing problems are studied for a class of distributed-parameter information and control systems described by linear partial differential or integro-differential equations. The results are first derived in discrete-time form following a Bayesian information processing approach and are then converted in continuous-time form using an extension of Kalman's limiting procedure. The paper concludes by showing how the results should be formally used for treating nonlinear systems. The filters derived here are the most general and involve the distributed-parameter filters derived by other authors as special cases.

Linear Regression Filtering and Prediction for Tracking Maneuvering Aircraft Targets

Abstract: A Cartesian coordinate linear regression filter is utilized for tracking maneuvering aircraft targets. Measurements of target position are made in a line-of-sight coordinate frame, but filtering is performed in Cartesian coordinates. Numerical results are given for optimizing the truncation time constant such that a good balance is obtained between the dynamic errors and the standard deviations. Lower bounds on the dynamic errors are established for the Cartesian coordinate linear regression filter and compared with a line-of-sight coordinate Kalman filter.

On the adaptive control of linear systems using the open-loop-feedback-optimal approach

Abstract: This paper considers the suboptimal stochastic control of linear discrete-time dynamical systems with unknown or stochastically varying parameters. The sub-optimal scheme is based upon the use of the open-loop-feedback-optimal (O.L.F.O.) method. The state and parameter estimates are generated by an extended Kalman filter algorithm. Numerical results for first order systems are presented.

Decision-directed adaptive recursive estimators: Divergence prevention

Abstract: In recent years, minimum-variance recursive estimators, such as the Kalman filter, have been used successfully in many practical applications. However, a common problem, known in the literature as the divergence phenomenon, is often encountered in these applications. Divergence is said to occur when the error covariance calculated by the estimator becomes inconsistent with the actual error covariance. Previous methods of dealing with this problem have involved limited memory filtering or simultaneous estimation of random process statistics. The method presented here is different in that the state model and statistics are accepted as given; the form of the optimal estimator is used, but a constant check is made on the consistency of the calculated and actual error covariances. The method is independent of the source of error, whether it be inaccuracies in the system model, incorrect values of the a priori and random process statistics, approximations required in the case of nonlinear systems, or computational roundoff. A test for inconsistency and an adaptive decision-directed procedure for adjusting the calculated covariance, shown to be optimal in a certain sense, are discussed. Several simulated examples, in which inconsistencies in the calculated and actual error covariances exist, show a significant improvement in the performance of the estimator when the given procedure is appended.

Reduced-order observers for linear discrete-time systems

Abstract: Luenberger's observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. An interesting new solution to the problem of constructing optimal and suboptimal reduced-order observers is presented. The solution contains as special cases both Kalman's optimal filter and the optimal minimal-order observer of Leondes and Novak. Also, the Tse and Athans observer is obtained as a special case of the reduced-order observer solution.

Pseudo State Measurements Applied to Recursive Nonlinear Filtering

Abstract: : Pseudo state measurements are constructed to make the measurement (geometry) model linear in the state. In the past, linear measurements have often proved to give better state estimates than nonlinear measurements. They are nonlinear functions of the actual measurement model bias parameters and are constructed to be linear functions of the state variables or to vanish in the absence of model or measurement error. Some examples of constructing pseudo state measurements are given in the paper. Recursive filter equations are derived using the pseudo state measurements and including colored (Markov) measurement noise and unestimated state and measurement model parameters. The filter estimates minimize the usual weighted least squares cost function with correlated state and pseudo state measurements. The filter is linear by construction. Higher order partial derivatives, if retained, would appear only in the computation of error variance and covariance matrices.

Approximate estimation for systems with quantized data

Abstract: Estimation of the state of a nonlinear discrete-time system using quantized data is considered. An exact solution for the maximum likelihood estimate is expressed as the solution of a nonlinear two-point boundary-value problem. Approximate recursive solutions for both the maximum likelihood and the conditional-mean estimates are obtained. The results of Monte-Carlo simulations are presented in which the performance of these two algorithms is compared with that of a Kalman filter in which the quantization error is approximated by white noise.

A simple method for the control of divergence in Kalman-filter algorithms

Abstract: It is found that, under certain conditions, the estimation errors produced by the Standard Kalman-filter algorithm increase rapidly, and become unbounded, even though the predicted error covariance continues to decrease in accordance with the stability properties of the Kalman filter. A very simple modification, which freezes the filter gain when divergence is suspected, is suggested. The modified algorithm would keep these errors within bound without causing an appreciable increase in the computation burden.

Comments on "On-line identification of linear dynamic systems with applications to Kalman filtering"

Abstract: The approach to discrete system identification described in a recent paper by Mehra is shown to be one example of a whole class of instrumental variable (IV) solutions. A recursive version of this IV solution is presented and an alternative, statistically more efficient, approximate maximum likelihood procedure is outlined.

Triangular Decomposition of a Positive Definite Matrix Plus a Symmetric Dyad with Applications to Kalman Filtering

Abstract: : An algorithm for the triangular decomposition of the sum of a positive definite matrix and a symmetric dyad is described. Several applications of the algorithm to the implementation of a square root Kalman filter are given.

Closed-loop gun control system

Abstract: A method of directing the firing of a gun and a system for its implementation. A final gun-fire order, aim correction, is deduced analytically based on observed performance, for example the miss distance of at least one previously fired projectile. Both the correlated and uncorrelated aspects as well as the bias of the observed performance are accounted for by assuming that the correlated aspects may be represented as a Markov process. A Kalman filter, linear predictor, and a storage-feedback device can then be employed to derive a correction order which may be applied to a subsequently fired projectile, thus minimizing the mean square miss distance.

Optimal inputs for system identification

Abstract: Identification criteria are presented for linear dynamic systems with and without process noise. With process noise, the state equations are replaced by the Kalman filter equations. If the identification performance index is expanded in a Taylor's series with respect to the parameters to be identified, then maximizing the weighting factor of the quadratic term with respect to the inputs will insure that an identification algorithm will converge more rapidly and to a more accurate result than with nonoptimal inputs. The expectation of this weighting factor is the Fisher information matrix, and its inverse is a lower bound for the covariance of the parameters. Direct and indirect methods of calculating the information matrix are presented for systems with and without process noise.

A Kalman filter as a minimax estimator

Abstract: A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.

Spectral Factorization in Periodically Time-Varying Systems and Application to Navigation Problems

Abstract: Spectral factorization is a powerful tool in deriving the steady-state solution of Kalman filtering equations. It is an algebraic, nortrecursive method, thus economical in terms of computing cost, when compared with the conventional iterative algorithm. In this paper the technique is extended to time-varying systems having periodic coefficient matrices for both discrete and continuous systems. The tracking of low-thrust spacecraft from an Earth-based station is used as an example and a sensitivity study is performed using a computer program incorporating the algorithm.

A New Approach to System Identification and State Estimation

Abstract: An effective on-line procedure is presented for system identification and state estimation for multiple-input multiple-output linear systems. The proposed solution offers consistent system identification, bounding of the steady-state covariance, and computational savings. Also, this solution is applicable to a larger class of problems than previously proposed solutions.

An Optimal Control Approach to Designing Constant Gain Filters

Abstract: This paper applies optimal control theory to designing constant gain filters which minimize a weighted average of the filtered variances. Uniaxial second-order motion is studied in detail, and an example is given which indicates that a constant gain filter may be designed with performance comparable to a Kalman filter. An appendix is included which shows how the approach may be extended to higher order systems.

An Adaptive Estimator for Passive Range and Depth Determination of a Maneuvering Target

Abstract: : The report describes an adaptive state estimator that can significantly improve the passive range and depth determination of a randomly maneuvering target. The target in this study is a submarine, which, while being tracked, performs large-magnitude depth changes at times unknown to the tracking submarine. Present passive tracking techniques usually utilize a Kalman filter to process the azimuth and/or elevation observations. A Kalman filter will theoretically give the 'best' estimates of target range, depth, and velocity when the system and measurement errors can be modeled as Gaussian processes. The main difficulty in using a Kalman filter in passive tracking applications is that large bias errors invariably develop as the target makes large alterations in velocity or depth. A technique for including a feedback-type learning processor in conjunction with the Kalman filter has been found to greatly reduce bias errors produced by the maneuvering target.