Showing papers on "Invariant extended Kalman filter published in 1976"
01 Dec 1976
TL;DR: The purpose of this paper is to examine several Kalman filter algorithms that can be used for state estimation with a multiple sensor system and the data compression method is shown to be computationally most efficient.
Abstract: The purpose of this paper is to examine several Kalman filter algorithms that can be used for state estimation with a multiple sensor system. In a synchronous data collection system, the statistically independent data blocks can be processed in parallel or sequentially, or similar data can be compressed before processing; in the linear case these three filter types are optimum and their results are identical. When measurements from each sensor are statistically independent, the data compression method is shown to be computationally most efficient, followed by the sequential processing; the parallel processing is least efficient.
245 citations
TL;DR: In this article, a simple suboptimal parameter and state estimator is presented which fills the need for economical, robust parameter-state estimators for adaptive controllers using minicomputers.
Abstract: The practical implementation of adaptive controllers using minicomputers requires algorithms which are both numerically economical and robust. The problem of combined state and parameter estimation for adaptive controllers was originally posed as a nonlinear filtering problem. The only known nonlinear filter which can be practically implemented on a small computer is the extended Kalman filter. The extended Kalman filter, however, often diverges, thus, there is a need for economical, robust parameter-state estimators. A simple suboptimal parameter and state estimator is presented which fills this need. The filter is based on a particular canonical form for the state-space equations of a linear system which allows the parameters and states to be estimated separately using two linear estimators. If an innovations model is used, the steady-state Kalman filter gains can be estimated and thus, during steady-state operation, the estimates of the states can be easily obtained. Numerical exampies are presented which demonstrate the increased robustness and speed of the proposed linear estimator over the extended Kalman filter.
144 citations
TL;DR: Empirical justification is established for the common practice of applying the Kalman filter estimator to three classes of linear quadratic problems where the model statistics are not completely known, and hence specification of the filter gains is not optimum.
Abstract: In this paper, theoretical justification is established for the common practice of applying the Kalman filter estimator to three classes of linear quadratic problems where the model statistics are not completely known, and hence specification of the filter gains is not optimum. The Kalman filter is shown to be a minimax estimator for one class of problems and to satisfy a saddlepoint condition in the other two classes of problems. Equations for the worst case covariance matrices are given which allow the specifications of the minimax Kalman filter gains and the worst case distributions for the respective classes of problems. Both time-varying and time-invariant systems are treated.
95 citations
11 Feb 1976
33 citations
TL;DR: In this article, a new method is proposed based on the quadratic criterion which evaluates an equation error, not an output error, which leads to a linear problem whereas the output error leads to nonlinear problem.
Abstract: This correspondence considers the problem of modeling linear time-invariant systems by linear systems of low order. A new method is proposed here based on the quadratic criterion which evaluates an equation error, not an output error. The minimization of an equation error leads to a linear problem whereas the output error leads to a nonlinear problem.
31 citations
TL;DR: In this article, a trajectory sensitivity approach is taken to the design of a Kalman filter for a system with uncertain parameters, where the performance index depends on the variances of the parameter deviations.
Abstract: A trajectory sensitivity approach is taken to the design of a Kalman filter for a system with uncertain parameters. A two-point boundary value problem (TPBVP) is formulated, where the performance index depends on the variances of the parameter deviations. A suboptimal algorithm is also developed. An example shows that estimation errors may be reduced considerably from those generated by a Kalman filter design for the nominal parameter values.
30 citations
14 Apr 1976
TL;DR: Several Kalman filter algorithms that can be used for state estimation with a multiple sensor system are examined and their results are compared with a suboptimum tracking algorithm which processes only multiple range measurements.
Abstract: : The purpose of this report is to examine several Kalman filter algorithms that can be used for state estimation with a multiple sensor system. These algorithms are described in detail and their results are compared with a suboptimum tracking algorithm which processes only multiple range measurements. A state estimate compression algorithm is also described. Various radar measurement transformation formulas are listed. Algorithms for a nonsynchronous data collection system are not examined in detail but possible approaches are suggested.
29 citations
01 Jan 1976
TL;DR: The positive result of this study is that the UD covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs.
Abstract: Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms, Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the UD covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the UD filter is further demonstrated by its insensitivity to variations in the a priori statistics.
15 citations
TL;DR: In this article, the Kalman filter has been extended to continuous-time linear dynamic systems and has gained popularity because the equations describing the filter are mechanized without the necessity of understanding the underlying theory of optimal estimation.
Abstract: Publisher Summary The advent of the modern digital computer opened a new realm of approaches to the problem of optimal estimation. One of the widely used digital computer oriented methods was formulated by Kalman. The technique provides a minimum-variance unbiased estimate of the state of a discrete-time linear dynamic system whose input and output are corrupted by additive Gaussian white noise. The approach was extended to continuous-time linear dynamic systems. The estimation technique has acquired the name “Kalman filter.” The Kalman filter has gained popularity because the equations describing the filter are mechanized without the necessity of understanding the underlying theory of optimal estimation. The Kalman filter has an appeal to control system engineers because its operation could be represented as a feedback control system. The drawback of Kalman filter is that the optimal filter equations require an exact knowledge of the system's dynamical equations and the statistics of the random quantities.
11 citations
01 Dec 1976
TL;DR: In this article, an unbiased minimum variance estimator which is constrained to be recursive is presented. But the form of the estimator is linear in terms of the innovation, although the gain does depend on past measurements.
Abstract: An algorithm is described which estimates the state of a linear system from quantized measurements of the output of that system. The estimator is an unbiased minimum variance estimator which is constrained to be recursive. The form of the estimator is linear in terms of the innovation, although the gain does depend on past measurements. The basic estimator is a time-varying filter; a stationary estimator, whose gain is computed prior to the processing of any measurements, is also presented. The performance of this quantized data filter is compared with the performances of both a Kalman filter operating on the linear output and a Kalman filter which processes quantized data. The quantized data filter results in significant performance improvements when a coarse quantization characteristic with few levels is used.
10 citations
TL;DR: A critical study of the measures of filter quality and the control actions which these methods employ is presented and a new method is suggested for control of divergence in the standard filter.
Abstract: Techniques for control of divergence of the Kalman filter are considered. The application in practice of three decision-directed methods for divergence prevention is investigated. A critical study of the measures of filter quality and the control actions which these methods employ is presented and a new method is suggested for control of divergence in the standard filter.
24 Mar 1976
TL;DR: The reduced update Kalman filter is derived and is shown to be optimum in that it minimizes the post update mean square error under the constraint of updating only the nearby previously processed neighbors.
Abstract: The Kalman filtering method is extended to two-dimensions. The resulting computational load is found to be excessive. The reduced update Kalman filter is derived. It is shown to be optimum in that it minimizes the post update mean square error (mse) under the constraint of updating only the nearby previously processed neighbors. The resulting filter is a stable, nonsymmetric half-plane recursive filter. This method is proposed as a solution of the 2-D filter design problem for stochastic dynamical models.
TL;DR: In this paper, the first and second nonstationary moments on the state, state noise, and measurement noise in a discrete-time, linear, dynamic stochastic system are identified.
Abstract: Publisher Summary Least squares estimation techniques are employed to identify the first and second non-stationary moments on the state, state noise; and measurement noise in a discrete-time, linear, dynamic stochastic system. The more accurately these statistics are known, the more accurate are the state estimates of a Kalman filter applied to this system. Least squares estimates are obtained of the original state, the means, and the covariance parameters without the necessity of specifying the distributions on the noise of any of the systems. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied for each system. The motivation for estimating the system statistics is to achieve accurate and rapidly converging estimates of the state of the system with a Kalman filter. When the first two moments are known, the Kalman filter produces accurate estimates of the state than any other linear filter.
TL;DR: For a set of stable, two-dimensional models with an uncertainty in the form of a constant, unknown control input, it is observed that any age-weighting of the data can cause an error greater than the Kalman filter.
Abstract: The steady-state error of the fading memory filter is studied for stable, two-dimensional models with an uncertainty in the form of a constant, unknown control input. For a set of these models, it is observed that any age-weighting of the data can cause an error greater than that of the Kalman filter. This contradicts previous assumptions about the fading memory filter.
TL;DR: In this article, the error analysis of the Schmidt-Kalman filter message and observation models with uncertain parameters is studied. But the error quantity considered is the actual covariance which is the mean square of the difference between the nominal state and the misidentified estimate.
Abstract: This paper deals with the error analysis of the Schmidt—Kalman filter message and observation models.which contain uncertain parameters. The error quantity considered is the actual covariance which is the mean square of the difference between the nominal state and the misidentified estimate. After deriving the Schmidt—Kalman filter, the error analysis is considered due to two causes, one of which is misidentifying the coefficients of the system, the covariances of the noises and the variance of the initial state and the other of which is simplifying the coloured measurement noise by the white measurement noise. A boundedness theorem of the error equation is considered in the remaining part. Only systems governed by continuous-time linear equations are treated here.
01 Dec 1976
TL;DR: In this paper, an adaptive extended Kalman filter (EKF) is used in an adaptive laser Optics Techniques (ALOT) control loop to track the glint from a spherical target.
Abstract: : An adaptive extended Kalman filter (EKF) is used in an Adaptive Laser Optics Techniques (ALOT) control loop to track the glint from a spherical target. A fourfold improvement in tracking bandwidth is obtained over a conventional conical scan ALOT tracking loop implemented with the same hardware. The increase in tracking bandwidth is manifested in the ratio between the (conical scan) dither frequency and the tracking bandwidth which is 2.5:1 for the EKF and 10:1 for the conventional analog or digital glint tracking schemes. The EKF provides correct pointing error estimates over essentially the entire range of the pointing error, whereas in a conventional conical scan loop the demodulated error versus true pointing error function is highly non-linear due to the nonlinearity of the glint.
14 Dec 1976
TL;DR: In this paper, an interactive scheme based on the Mahalanobis distance function is used to detect target maneuvers, and an interactive approach is proposed to increase the error covariance matrix to its proper value.
Abstract: : The Kalman filter is the optimum tracking filter regardless of whether or not radar detections are made from single or multiple platforms. The performance of the Kalman filter has been simulated for various radar-target geometries. An error criterion involving the Mahalanobis distance function is used to detect target maneuvers, and an interactive scheme based on this criterion is used to increase the error covariance matrix to its proper value. Attempts to replace the Kalman filter with a simple filter with comparable performance have not been productive. The basic reason behind this difficulty is that accurate position and velocity estimates (obtainable by triangulation from different platforms) require the processing of position and velocity covariance matrices. Since both matrices must be saved and updated, a simple filter does not seem possible.
TL;DR: In this article, a new derivation of continuous-time Kalman Filter equations is presented, and a unified approach to filtering and smoothing problems has thus been achieved, which has been previously used to derive the smoothing equations.
01 Dec 1976
TL;DR: In this article, a large number of ad hoc modifications are required to prevent divergence, resulting in a rather complex filter and performance is quite good as judged by comparison of Monte-Carlo simulations with the Cramer-Rao lower bound, and by the filter's ability to track maneuvering targets.
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 filter's ability to track maneuvering targets.
16 Aug 1976
TL;DR: In this article, the development of an electro-optical mercury tiltmeter, its use to monitor low frequency motions, and an iterated extended Kalman filter to enhance the knowledge of the tilt output by filtering are described.
Abstract: Testing a high quality inertial sensor requires a precise knowledge of its motion environment. This presents a considerable problem for state-of-the-art instruments which are capable of sensing motions as small as a few thousandths of an arc second. This paper describes the development of an electro-optical mercury tiltmeter, its use to monitor low frequency motions, and the development of an iterated extended Kalman filter to enhance the knowledge of the tilt output by filtering.
TL;DR: In this paper, the duality relation between the actual covariance equation of the Kalman filter and the matrix equation caused by a performance loss of the non-singular control problem was investigated.
Abstract: Error analysis for the two kinds of low-order filters is investigated. One low-order filter is an optimal type and the other filter is a sub-optimal Luenberger observer type. As the error analysis, algorithms representing the two actual covariances and the related moments are derived from the nominal dynamical systems and the actual low-order filters whose parameters are misidentified. It is also clarified that the duality relation does not exist between the actual-Covariance equation of an optimal low-order filter and the matrix equation of a sub-optimal performance value of the singular control problem. This investigation is motivated by the result that the duality relation exists between the actual-covariance equation of the Kalman filter and the matrix equation caused by a performance loss of the non-singular control problem.
TL;DR: In this paper, a well-posed optimisation problem is formulated using a quadratic error index where incremental corrections ΔQ and ΔR are obtained at the end of each of progressive overlapping observation intervals.
Abstract: In the present paper a well-posed optimisation problem is formulated using a quadratic error index where incremental corrections ΔQ and ΔR are obtained at the end of each of progressive overlapping observation intervals. For each new obtained value of Q and R a new value for the steady-state optimum Kalman filter gain is computed. At the end of the optimisation procedure a specific Kalman filter is obtained where the resulting constant gain is the correct steady-state optimum Kalman filter gain. When comparing the developed algorithm with an alternative one where the constant filter gain is computed directly without identifying the unknown noise covariances, the results show the sensitivity of the latter method to the initial guesses of the filter gain, a difficulty that is not existent in the present algorithm.
01 Dec 1976
TL;DR: In this article, the problem of estimating the state variables and maneuvering parameters for a re-entry vehicle was considered and non-linear filtering results based upon a tuned nine state extended Kalman filter were compared with those obtained by a linearized fixed interval smoother (for non-realtime applications) using Monte Carlo simulation experiments.
Abstract: This paper considers the problem of estimating the state variables and maneuvering parameters for a re-entry vehicle. The non-linear filtering results based upon a tuned nine state extended Kalman filter (for realtime applications) are compared with those obtained by a linearized fixed interval smoother (for non-realtime applications) using Monte Carlo simulation experiments.