Showing papers on "Alpha beta 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: In this article, a theory of Luenberger type of observers for time-delay systems based on spectral decomposition techniques is developed and the stability, of the observer is rigorously established.
Abstract: A theory of Luenberger type of observers for time-delay systems based on spectral decomposition techniques is developed. The stability, of the observer is rigorously established and the observer equations are reduced to an integro-differential form. The computational aspects are discussed and the difficulty is traced to that of computing eigenvalues of delay differential equations. Formal structural similarities of Kalman filters for delay systems and the observer developed here are indicated.
138 citations
TL;DR: In this paper, a minimal order state observer for a bilinear system is considered and the necessary condition for the existence of such an observer and a standard form of the observer satisfying this condition is presented.
Abstract: This paper considers a minimal order state observer for a bilinear system. The given observer is an extension of one for a linear system and can be designed without considering inputs because the estimation error is made to be independent of inputs. The necessary condition for the existence of a minimal order observer and a standard form of a bilinear system satisfying this condition are presented. A standard form of a minimal order state observer is also obtained and the design procedure of this observer is shown. As an example, the observer of a d.c. motor is designed.
124 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
01 Feb 1976
TL;DR: A model for the human observer in failure detection tasks is proposed which consists of a linear estimator and a decision mechanism which leads to a decision function which is the integration of the filter residuals.
Abstract: A model for the human observer in failure detection tasks is proposed which consists of two stages: a linear estimator and a decision mechanism The estimator is a Kalman filter, and the decision mechanism, which is based on Wald's sequential analysis, leads to a decision function which is the integration of the filter residuals The final result is a simple detection system which depends on only three parameters, and the sensitivity of the model to these parameters is analyzed The results of an experiment designed to test the validity of the model are reported The question of open and closed decision intervals as well as the generalization of the model to more complicated cases is discussed
74 citations
TL;DR: In this paper, the tracking of a moving source (MS) in shallow water is analyzed, where the observations of bearing, β, and any two time delays, τ 1 and τ 2, are made by a single observer, as opposed to spatially separated observers.
50 citations
11 Feb 1976
33 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
TL;DR: Compatibility of the observer designed for a linear time-invariant system is discussed and a class of systems to which a given observer is compatible is explicitly stated.
Abstract: Compatibility of the observer designed for a linear time-invariant system is discussed and a class of systems to which a given observer is compatible is explicitly stated. This class gives allowable variations of system parameters such that the designed observer is still compatible to the system when the system changes its characteristics. An application to the regulator problem is described briefly.
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.
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.
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.
16 Aug 1976
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 article, the problem of designing an observer to estimate a vector-valued (multi-) linear function of the state of a linear time-invariant system, for the purpose of implementing a feedback control law is discussed.
Abstract: This paper discusses the problem of designing an observer to estimate a vector-valued (multi-) linear function of the state of a linear time-invariant system, for the purpose of implementing a feedback control law. First, the standard form of an observer is obtained by using the Luenberger observable canonical form. Second, the design procedure of a multi-linear functional observer is derived using this standard form. Observers derived using this procedure are generally o flower order than those of Luenberger. Lastly, the procedure presented is illustrated by an example.
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 paper, the status of observer theory and results obtained in stochastic observer theory as applied to discrete-time linear systems are discussed and a method of constructing a reduced-order observer estimator is presented.
Abstract: Publisher Summary This chapter reviews the status of observer theory and presents the results obtained in stochastic observer theory as applied to discrete-time linear systems. The Kalman filter solves the problem of state estimation in the mean-square sense for linear discrete-time stochastic systems, numerical and computational problems associated with the real-time implementation of Kalman filters have led researchers to seek out computationally simpler solutions to the minimum mean-square state-estimation problem. The chapter reviews some of the alternate approaches to the discrete-time state-estimation problem based on the extension of Luenberger's observer theory to stochastic systems. The chapter discusses the notion of an observer for discrete stochastic systems. It presents a method of constructing a reduced-order observer estimator. The case of some perfect measurements is considered. The interpretations of Brammer's optimal observer, namely, a Kalman-type algorithm and a Luenberger-type algorithm have been presented. Computational advantages of the reduced-order observer algorithm have been presented.
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.
TL;DR: In this article, a new adaptive observer is proposed which combines the advantages of the two types of observers described above, and the parameter estimates of the plant are directly obtained with a structure which is no more complex than that of a non-minimal observer which is widely used at the present time.
Abstract: : An adaptive observer is defined as one which estimates the state variables and parameters of an unknown stable linear time-invarient plant from its input-output data. At the present time, there are two distinct approaches to the design of adaptive observers for a plant whose input output behavior can be represented by an n-th order differential equation. In the first approach, the observer is of the same order as the plant and is referred to as a minimal (order) observer. Using the second approach, a non-minimal observer of order (2n-1) is obtained. Minimal observers are considerably more difficult to synthesize than non-minimal observers and require the generation of additional signals for the stabilization of the adaptive loop. However, they have the advantage of yielding simultaneously both parameter and state estimates of the plant. Non-minimal observers are considerably simpler in structure both the n state variables of the plant have to be estimated from the available (2n-1) state variables of the observer. In this brief paper, a new observer is proposed which appears to combine the advantages of the two types of observers described above, With this observer, the parameter estimates of the plant are directly obtained with a structure which is no more complex than that of a non-minimal observer which is widely used at the present time. The parameter estimates are simultaneously used to determine directly the state estimates of the plant. Under certain conditions, the new observer has a faster rate of convergence than the observers known at present, which makes it particularly attractive for use in the control problem.
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 Nov 1976
TL;DR: In this article, the problem of designing an optimal filter for a beta gauge is considered, and a simple approximation to the prefiltered observation process is shown to be adequate in cases where the Poisson pulse rate is high.
Abstract: The paper considers the problem of designing an optimal filter for a beta gauge. The output from this kind of gauge is a doubly stochastic Poisson process, and, for the linear problem, the optimal filter is shown to have a particularly simple form. If the gauge output pulse train is not available directly, but only in a prefiltered form, then an optimal filter can still be developed, although it is difficult to implement. A simple approximation to the prefiltered observation process is shown to be adequate in cases where the Poisson pulse rate is high. In the linear case, this allows the well known Kalman-Bucy filter to be used. A numerical example demonstrates the usefulness of the approximate filter and confirms the conclusion that in practice the Kalman-Bucy filter is more than adequate.