Showing papers on "Kalman filter published in 1977"

Robust bayesian estimation for the linear model and robustifying the Kalman filter

Abstract: Starting with the vector observation model y = Hx + v , robust Bayesian estimates \hat{x} of the vector x are constructed for the following two distinct situations: 1) the state x is Gaussian and the observation error v is (heavy-tailed) non-Gaussian and 2) the state is heavy-tailed non-Gaussian and the observation error is Gaussian. Bounds with respect to broad symmetric non-Gaussian families are derived for the error covariance matrix of these estimates. These "one-step" robust estimates are then used to obtain robust estimates for the Kalman filter setup y_{k}= H_{k}x_{k}+ v_{k}, x_{k+1}=\Phi_{k}x_{k}+w_{k} . Monte Carlo results demonstrate the robustness of the proposed estimation procedure, which might be termed a robustified Kalman filter.

Application of Fast Kalman Estimation to Adaptive Equalization

Abstract: Very rapid initial convergence of the equalizer tap coefficients is a requirement of many data communication systems which employ adaptive equalizers to minimize intersymbol interference. As shown in recent papers by Godard, and by Gitlin and Magee, a recursive least squares estimation algorithm, which is a special case of the Kalman estimation algorithm, is applicable to the estimation of the optimal (minimum MSE) set of tap coefficients. It was furthermore shown to yield much faster equalizer convergence than that achieved by the simple estimated gradient algorithm, especially for severely distorted channels. We show how certain "fast recursive estimation" techniques, originally introduced by Morf and Ljung, can be adapted to the equalizer adjustment problem, resulting in the same fast convergence as the conventional Kalman implementation, but with far fewer operations per iteration (proportional to the number of equalizer taps, rather than the square of the number of equalizer taps). These fast algorithms, applicable to both linear and decision feedback equalizers, exploit a certain shift-invariance property of successive equalizer contents. The rapid convergence properties of the "fast Kalman" adaptation algorithm are confirmed by simulation.

White-noise estimators for seismic data processing in oil exploration

Abstract: This paper is motivated by a problem from seismic data processing in oil exploration. We develop a Kalman filtering approach to obtaining optimal smoothed estimates of the so-called reflection coefficient sequence. This sequence contains important information about subsurface geometry. Our problem is shown to be equivalent to that of estimating white-plant noise for a linear dynamic system. By means of the equations which are derived herein, it is possible to compute fixed-interval, fixed-point, or fixed-lag optimal smoothed estimates of the reflection coefficient sequence, as well as respective error covariance-matrix information. Our optimal estimators are compared with an ad hoc "prediction error filter," (PEF) which has recently been reported on in the geophysics literature. We show that one of our estimators performs at least as well as, and in most cases, better than the prediction error filter. Simulation results are given which support and illustrate the theoretical developments.

On the state and parameter estimation for maneuvering reentry vehicles

Abstract: This paper examines several versions of the extended Kalman filter which can be used to estimate the position, velocity, and other key parameters associated with maneuvering reentry vehicles. These filters are discussed in terms of the fundamental problems of modeling accuracy, filter sophistication, and the real-time computational requirements. Techniques which adaptively increase the process noise to compensate for modeling errors during the maneuvers are examined.

A tracking filter for maneuvering sources

Abstract: It is well known that the extended Kalman filtering methodology works well in situations characterized by a high signal-to-noise ratio, good observability and a valid state trajectory for linearization. This paper considers a problem not characterized by these favorable conditions. A large number of ad hoc modifications are required to prevent divergence, resulting in a rather complex filter. However, performance is quite good as judged by comparison of Monte Carlo simulations with the Cramer-Rao lower bound, and by the filters ability to track maneuvering targets.

New Fast Optimal White-Noise Estimators for Deconvolution

Abstract: We present a brief qualitative discussion of Kalman filtering as contrasted with Wiener filtering, since the Kalman filter is an integral element in our new fast optimal white-noise estimators. Additionally, we present two fast algorithms, one of which is shown to be very efficient for calculating fixed-interval estimates of the reflection coefficient sequence, the other of which is shown to be very efficient for calculating either fixed-point or fixed-lag estimates of that sequence. Detailed operation counts are given which support these claims. Flow charts are also given for the Kalman filter and the two new fast smoothing algorithms.

Kalman Filtering Applied to Statistical Forecasting

Abstract: This paper describes the use of the Kalman Filter in a certain ciass of forecasting problems. The time series is assumed to be modeled as a time varying mean with additive noise. The mean of the time series is assumed to be a linear combination of known functions. The coefficients appearing in the linear combination are unknown. Under such assumptions, the time series can be described as a linear system with the state vector of the system being the unknown parameters and present value of the mean of the process. The Kalman Filter can be used under these circumstances to obtain an “optimal” estimate of the state vector. One of the distinct advantages of the Kalman Filter is that time varying coefficients can be permitted in the model. Examples using the Kalman Filter in forecasting are presented.

Estimation theory

Abstract: Statistical Inference – methods by which generalizations are made about a population Two Major Areas of Statistical Inference 1. Estimation – a parameter is established based on the sampling distribution of a proportion, establishing a certain degree of accuracy from the estimate 2. Test of Hypothesis – a decision is arrived at about a prestated hypothesis, thereby accepting or rejecting the hypothesis.

Compatibility check of measured aircraft responses using kinematic equations and extended Kalman filter

Abstract: An extended Kalman filter smoother and a fixed point smoother were used for estimation of the state variables in the six degree of freedom kinematic equations relating measured aircraft responses and for estimation of unknown constant bias and scale factor errors in measured data. The computing algorithm includes an analysis of residuals which can improve the filter performance and provide estimates of measurement noise characteristics for some aircraft output variables. The technique developed was demonstrated using simulated and real flight test data. Improved accuracy of measured data was obtained when the data were corrected for estimated bias errors.

Analysis of EEG signals with changing spectra using a short-word Kalman estimator

Abstract: A new method is described for the numerical analysis of electroencephalographic signals, whose spectral distributions vary with time. The variation need not be slow. The theory describes a sampled EEG by an autoregressive series of high order with stochastic coefficients having independent increments. The coefficients are augmented as state variables and can thus be estimated in real time by a special Kalman filter. This raises, however, two problems: (1) The Riccati equation depends on the signal and is numerically unstable. (2) Specifications for tuning the filter are unknown; the essential parameter is the rate of change of the stochastic coefficients. For solutions, a numerically stable algorithm is derived (not “square-root filtering”), and the rate of change is estimated according to maximum likelihood. This defines an “index of nonstationarity”, the value of which is fundamental for the analysis. Applications to recorded EEG signals have demonstrated the feasibility of the method.

A Semicausal Model for Recursive Filtering of Two-Dimensional Images

Abstract: A two-dimensional discrete stochastic model for representing images is developed. This representation has lower mean square error, compared to a standard autoregressive Markov representation. Application of the model to linear filtering of images degraded by white noise leads to scalar recursive filtering equations requiring only 0(N2log 2 N) computations for N x N images. The filter algorithm is a hybrid algorithm where the image is transformed along one dimension and spatially filtered, recursively, in the other. Examples on a 255 X 255 image are given.

A suboptimal estimation algorithm with probabilistic editing for false measurements with applications to target tracking with wake phenomena

Abstract: The purpose of this paper is to describe a suboptimal technique called probabilistic edit, which can be used for state variable estimation in conjunction with Kalman filtering techniques when the underlying noisy measurement process can contain false measurements, i.e., measuremernts containing no information about the state variables at certain random periods of time. The basic probabilistic edit algorithm is modified to accommodate real-time considerations and is incorporated into a seven-state extended kalman filter which: 1) tracks ballistic reentry vehicles, and 2) estimates their ballistic coefficient. In estimation problems of this kind, the radar measurements of the reentry vehicle's position are corrupted due to contamination of the hard body return with that of the wake. Consequently, a degradation in the performance of the basic tuned extended Kalman filter occurs. Thus, measurements that appear (in a probabillstic sense) to be highly contaminated by wake are modeled as false measurements. The paper includes a discussion of the effect of wake, a description of the basic tracking algoritim, and modifications of the basic tracking algorithm to compensate for the wake-corrupted measurements. Finally, the performance of three distinct algorithm, 1) unmodified ballistic tracking filter, 2) modified ballistic tracking filter using a chi-square test to reject bad measurements, and 3) modified ballistic tracking filter using the probabilistic edit algorithm, using actual data will be presented. The comparison of the three algorithms is conducted using several distinct experiments which differ markedly in the percentage of measurements containing significant wake-corrupted data. The results indicate that the probabilistic edit algorithm can provide improved estimation of the ballistic coefficient with minimal additional real-time computational requirements.

Sensory Mechanism Modeling.

Abstract: : Pilots use information from a variety of sensory mechanisms to determine their estimate of orientation and motion. An understanding of this process and a quantitative model are essential for development of effective simulator motion cueing devices. A multisensory model for dynamic spatial orientation is being developed for this purpose. Aircraft or simulator motion is translated into stimuli which are processed by dynamic models of the appropriate sensors (visual, vestibular, tactile, and proprioceptive), and are then fed to a central estimator which has been modeled as a linear optimal estimator, specifically a steady state Kalman Filter. In addition to the linear estimation process, some non-linear effects, such as the well documented delay in onset of visually induced motion, require non-linear additions to the model. Such additions have been kept to a minimum so as to retain the uniqueness and conceptual appeal of a linear optimization algorithm. The model has been implemented as a computer program and has predicted some of the important qualitative characteristics of human dynamic spatial orientation under combined wide field visual motion and platform motion. Several types of special tactile and proprioceptive cues are also being considered but have not been validated. The modeling effort has underscored the need for additional data in some areas and several experiments have been suggested to help fill these gaps. (Author)

Filter Stability for Stochastic Evolution Equations

Abstract: It is established that the Kalman filter associated with signal and observation processes defined through stochastic evolution equations is stable under very weak hypotheses; namely when appropriate stabilizability / detectability criteria hold. Thus in this general setting we obtain results as sharp as are available for processes taking values in finite dimensional linear spaces. The conditions are shown to be directly verifiable in certain important situations.

Square-root algorithms for the continuous-time linear least squares estimation problem

Abstract: We present a simple differential equation for the triangular square root of the state error variance of the continuous-time Kalman filter. Unlike earlier methods of Andrews, and Tapley and Choe, this algorithm does not explicitly involve any antisymmetric matrix in the differential equation for the square roots. The role of antisymmetric matrices is clarified: it is shown that they are just the generators of the orthogonal transformations that connect the various square roots; in the constant model case, a similar set of antisymmetric matrices appears inside the Chandrasekhar-type equations for the square roots of the derivative of the error variance. Several square-root algorithms for the smoothing problem are also presented and are related to some well-known smoothing approaches.

Recursive least squares smoothing of noise in images

Abstract: In the recent past considerable attention has been devoted to the application of Kalman filtering to smoothing out observation noise in image data. A generalization of the one-dimensional Kalman filter to two dimensions was earlier suggested by Habibi, but it has since been shown that this generalization is invalid since it does not preserve the optimality of the Kalman filter. A new method is proposed here that enables well-established Kalman-filter theory to yield a simple two-dimensional filter for images that can be modeled by two-dimensional wide-sense Markov (WSM) random fields.

Parameter estimation via the kalman filter

Abstract: A method for parameter estimation is presented using the Kalman filter with appropriate initial conditions. The filter solution is shown to approximate the minimum-norm weighted least-squares solution to any desired accuracy during all phases of estimation. Furthermore, the computations are identical for each measurement, irrespective of whether a minimal observable data set has been established. This procedure contrasts with other techniques for parameter estimation that require additional computation when the process is unobservable.

Reduced order state estimators for discrete-time stochastic systems

Abstract: A reduced order, least squares, state estimator is developed for linear discrete-time systems having both input disturbance noise and output measurement noise with no output being free of measurement noise. The order reduction is achieved by using a Luenberger observer in connection with some of the system outputs and a Kalman filter to estimate the state of the Luenberger observer. The order of the resulting state estimator is reduced from the order of the usual Kalman filter system state estimator by the number of system outputs selected for use as inputs to the Luenberger Observer. The manner in which the noise associated with the selected system outputs affects the state estimation error covariance provides considerable insight into the compromise being attempted.

Failure detection by pilots during automatic landing: models and experiments

Abstract: A model is proposed to describe the pilot as a monitor of automatic landing systems. The failures treated are equivalent to the addition of a dynamic change in the mean of the observation process. The failure detection model of the pilot consists of two stages: a linear estimator (Kalman filter) and a decision mechanism based on sequential analysis. The filter equations are derived from a simplified version of the linearized dynamics of the airplane and the control loop. The perceptual observation noise is modified to include the effects of allocation of attention among the several instruments. The final result is a simple model consisting of a high-pass filter to produce the observation residuals and a decision function which is a pure integration of the residuals minus a bias term. The dynamics of a Boeing 707 were used to simulate the fully coupled final approach in a fixed-base simulator. Observers monitored the approaches and detected the failures; their performance was compared with the predictions of the model.

On bootstrap identification using stochastic approximation

Abstract: A two-stage state and parameter estimation algorithm for linear systems has been developed. Stage 1 uses a stochastic approximation method for state estimation, while stage 2 considers parameter estimation through a linear Kalman filter. These two stages are coupled in a bootstrap manner. The algorithm is computationally much simpler than the usual extended Kalman filter. A fourth-order numerical example has been solved, and results have been compared with those obtained using an extended Kalman filter.

Adaptive state estimation for systems with unknown noise covariances

Abstract: An adaptive scheme is proposed for obtaining the steady-state Kalman gain matrix for o discrete-time system without a priori knowledge of the noise covariance matrices. It is based on combining an algorithm proposed recently by Carew and Belanger with an algorithm based on stochastic approximation. Results of simulation are given comparing the proposed method with earlier algorithms.

Hierarchic recursive image enhancement

Abstract: The problem that is solved in this paper can be formulated as: given an observation of an image against the background of additive noise and given the statistics of the image and the noise, find an optimal estimate of the image such that the computer-time and storage requirements of the estimator are modest for images of, say 250 \times 250 points or more. A discrete-time vector-scanning model is derived that describes the statistics of a large class of images. The optimal linear smoother-with regard to the least-squares criterion-is formulated in a recursive manner as a combination of two Kalman filters. It is observed that in the model the covariance matrices are Toeplitz matrices. It is shown that the z transform defines a one-to-one relation between Toeplitz matrices and functions of a complex variable. This reduces the Riccati equation to a scalar equation in the z domain. It is further shown that multiplication by a Toeplitz matrix can be performed recursively by two linear dynamical systems. This leads to an algorithm which is not only recursive in the "time" parameter of the state space model but also in the index of the elements of the state vector. This so-called hierarchic recursive method has modest computational requirements.

Information and distortion in reduced-order filter design

Abstract: The relation and practical relevance of information theory to the filtering problem has long been an open question. The design, evaluation, and comparison of (suboptimal) reduced-order filters by information methods is considered. First, the differences and similarities between the information theory problem and the filtering problem are delineated. Then, based on these considerations, a formulation that {\em realistically} imbeds the reduced-order filter problem in an information-theoretic framework is presented. This formulation includes a "constrained" version of the rate-distortion function. The Shannon lower bound is used both to derive formulas for (achievable) rose lower bounds for suboptimal filters and to prove that for thc reduced-order filter problem the given formulation specifies a useful relation between information and distortion in filtering. Theorems addressed to reduced-order filter design, evaluation, and comparison based on information are given. A two step design procedure is outlined which results in a decoupling of thc search in filter parameter space, and hence in computational savings.

Second-order observer for nonlinear systems from discrete noiseless measurements

Abstract: This paper presents a second-order observer which estimates the states of a nonlinear plant based on discrete deterministic measurements. The observer is obtained by applying the time-varying linear observer theory to the augmented linearized model which is obtained by replacing each quadratic term in the original system with new state variables. The gain of the observer is evaluated sequentially through a procedure similar to that in the Kalman filter. The comparison between the extended linear observer and this one is made through the computer simulation for two model systems of nonlinear type wherein one of them is an electric power system model. These simulation results indicate that the use of second-order observer leads to improved estimation performance.