Showing papers on "Kalman filter published in 1971"
TL;DR: A density approximation involving convex combinations of gaussian density functions is introduced and proposed as a meaningful way of circumventing the difficulties encountered in evaluating these relations and in using the resulting densities to determine specific estimation policies.
965 citations
TL;DR: In this article, the square root approach is proposed to solve the problem of discrete filtering in the absence of a state estimate and an error covariance matrix from stage to stage, which is equivalent algebraically to the conventional Kalman approach.
Abstract: The conventional Kalman approach to discrete filtering involves propagation of a state estimate and an error covariance matrix from stage to stage. Alternate recursive relationships have been developed to propagate a state estimate and a square root error covariance instead. Although equivalent algebraically to the conventional approach, the square root filters exhibit improved numerical characteristics, particularly in ill-conditioned problems. In this paper, current techniques in square root filtering are surveyed and related by applying a duality association. Four efficient square root implementations are suggested, and compared with three common conventional implementations in terms of computational complexity and precision. The square root computational burden should not exceed the conventional by more than 50 percent in most practical problems. An examination of numerical conditioning predicts that the square root approach can yield twice the effective precision of the conventional filter in ill-conditioned problems. This prediction is verified in two examples. The excellent numerical characteristics and reasonable computation requirements of the square root approach make it a viable alternative to the conventional filter in many applications, particularly when computer word length is limited, or the estimation problem is badly conditioned.
467 citations
TL;DR: The Kalman estimation technique is examined from the point of view of the asymptotic behavior of the errors, and both "true" and "apparent" divergence are demonstrated by a simple scalar system.
Abstract: The Kalman estimation technique is examined from the point of view of the asymptotic behavior of the errors in the estimates. It is shown that, under certain conditions, the mean-square errors may become unbounded with time, and that this divergence may or may not be correctable by increasing the intensity of process noise assumed in the filtering model General results are derived for multidimensional systems, and both "true" and "apparent" divergence are demonstrated by a simple scalar system. Divergence due to numerical inaccuracies is considered, and an example problem in orbital navigation is used to demonstrate divergence and its elimination.
383 citations
TL;DR: In this article, a correlation technique which identifies a system in its canonical form is presented, which is capable of being implemented on-line and can be used in conjunction with the Kalman filter.
Abstract: Kalman gave a set of recursive equations for estimating the state of a linear dynamic system. However, the Kalman filter requires a knowledge of all the system and noise parameters. Here it is assumed that all these parameters are unknown and therefore must be identified before use in the Kalman filter. A correlation technique which identifies a system in its canonical form is presented. The estimates are shown to be asymptotically normal, unbiased, and consistent. The scheme is capable of being implemented on-line and can be used in conjunction with the Kalman filter. A technique for more efficient estimation by using higher order correlations is also given. A recursive technique is given to determine the order of the system when the dimension of the system is unknown. The results are first derived for stationary processes and are then extended to nonstationary processes which are stationary in the q th increment. An application of the results to a practical problem is presented.
277 citations
TL;DR: In this paper, the basic principles of least squares estimation are introduced and applied to the solution of some filtering, prediction, and smoothing problems involving stochastic linear dynamic systems.
Abstract: In this tutorial paper the basic principles of least squares estimation are introduced and applied to the solution of some filtering, prediction, and smoothing problems involving stochastic linear dynamic systems. In particular, the paper includes derivations of the discrete-time and continuous-time Kalman filters and their prediction and smoothing counterparts, with remarks on the modifications that are necessary if the noise processes are colored and correlated. The examination of these state estimation problems is preceded by a derivation of both the unconstrained and the linear least squares estimator of one random vector in terms of another, and an examination of the properties of each, with particular attention to the case of jointly Gaussian vectors. The paper concludes with a discussion of the duality between least squares estimation problems and least squares optimal control problems.
216 citations
TL;DR: In this paper, the authors compared the performance of several non-linear filters for the real-time estimation of the trajectory of a reentry vehicle from its radar observations, including iterative-sequential filters, single-stage iteration filters, and second-order filters.
Abstract: This paper compares the performance of several non-linear filters for the real-time estimation of the trajectory of a reentry vehicle from its radar observations. In particular, it examines the effect of using two different coordinate systems on the relative accuracy of an extended Kalman filter. Other filters considered are iterative-sequential filters, single-stage iteration filters, and second-order filters. It is shown that a range-direction-cosine extended Kalman filter that uses the measurement coordinate system has less bias and less rms error than a Cartesian extended Kalman filter that uses the Cartesian coordinate system. This is due to the fact that the observations are linear in the range-direction-cosine coordinate system, but nonlinear in the Cartesian coordinate system. It is further shown that the performance of the Cartesian iterative-sequential filter that successively relinearizes the observations around their latest estimates and that of a range-direction-cosine extended Kalman filter are equivalent to first order. The use of a single-stage iteration to reduce the dynamic nonlinearity improves the accuracy of all the filters, but the improvement is very small, indicating that the dynamic nonlinearity is less significant than the measurement nonlinearity in reentry vehicle tracking under the assumed data rates and measurement accuracies. The comparison amongst the nonlinear filters is carried out using ten sets of observations on two typical trajectories. The performance of the filters is judged by their capability to eliminate the initial bias in the position and velocity estimates.
210 citations
TL;DR: It is shown that it is often better to process statistically independent measurements in more than one batch and then to use sequential processing than to process them together via simultaneous processing.
Abstract: How practical is a Kalman filter? One answer to this question is provided by the computational requirements for the filter. Computational requirements-computational time per cycle (iteration) and required storage-determine minimum sampling rates and computer memory size. These requirements are provided in this paper as functions of the dimensions of the important system matrices for a discrete Kalman filter. Two types of measurement processing are discussed: simultaneous and sequential. It is shown that it is often better to process statistically independent measurements in more than one batch and then to use sequential processing than to process them together via simultaneous processing.
125 citations
01 Dec 1971
TL;DR: This filter reduces to the Kalman filter in the limiting case of no correlation errors and infinite gate sizes, and provides substantially improved performance in environments wherein correlation uncertainties cannot be ignored.
Abstract: This paper develops a new optimal tracking filter that accounts for and minimizes the effects of correlation uncertainties in surveillance systems. This filter reduces to the Kalman filter in the limiting case of no correlation errors and infinite gate sizes, and provides substantially improved performance in environments wherein correlation uncertainties cannot be ignored.
96 citations
74 citations
TL;DR: In this article, a variable-dimension stage-invariant Kalman filter was developed and used to estimate the state of the system based on measurements at each individual machine of instantaneous field and armature current, field and voltage, and angular deviation of machine rotor shaft from a synchronous phasor reference.
Abstract: Estimation of the dynamic state of a power system is the first prerequisite for control and stability prediction under transient conditions. Since the magnetic flux linkages which characterize the instantaneous state of the machines and the system are not directly measurable, a variable-dimension stage-invariant Kalman filter was developed and used to estimate the state of the system based on measurements at each individual machine of instantaneous field and armature current, field and armature voltage, and angular deviation of machine rotor shaft from a synchronous phasor reference. Gaussian white noise in the measurements was assumed, and the filter provided a recursive near-optimal minimum variance estimate of the state of the nonlinear system.
44 citations
01 Feb 1971
TL;DR: In this article, a simple characterisation of the optimal stationary Kalman-Bucy filter is obtained in terms of the return-difference matrix for the associated feedback system, which leads to a physical interpretation of the mechanism by which signal and noise are separated, which could form the basis of an approach to filter design.
Abstract: A strikingly simple characterisation of the optimal stationary Kalman-Bucy filter is obtained in terms of the return-difference matrix for the associated feedback system. The spectral factorisation of the observation spectral-density matrix is shown to generate directly the appropriate return-difference matrix. This leads to a physical interpretation of the mechanism by which signal and noise are separated, which could form the basis of an approach to filter design.
01 Dec 1971
TL;DR: In this paper, a simple characterisation of the optimal stationary discrete Kalman filter is obtained in terms of the return-difference matrix for the associated feedback system, which leads to a proof of a necessary condition for optimality of discrete multivariable-feedback filter and regulator problems.
Abstract: A simple characterisation of the optimal stationary discrete Kalman filter is obtained in terms of the return-difference matrix for the associated feedback system. A ztransform spectral factorisation of the observation spectral-density matrix is developed from the discrete-time matrix Riccati equation, and is shown to generate directly the appropriate return-difference matrix. This leads to a proof of a necessary condition for optimality of discrete multivariable-feedback filter and regulator problems.
TL;DR: The application of linear optimal control and least square filtering theory to the design of a control system for the Saturn V launch vehicle is described in this paper, where an optimal linear control is derived from a quadratic performance index in ratios of these critical variables to their design limits.
Abstract: The application of linear optimal control and least square filtering theory to the design of a control system for the Saturn V launch vehicle is described. System performance is evaluated on the basis of the ratio of the standard deviations of structural load, engine deflection, and lateral velocity drift to their design limits. An optimal linear control is derived from a quadratic performance index in ratios of these critical variables to their design limits. It is shown that this essentially eliminates trial and error search for suitable quadratic index weighting coefficients. It is also shown that the standard deviations are only slightly degraded when feedback gains on actuator states and bending and slosh modes are eliminated, leaving feedback gains on only the three rigid-body states and feed-forward gains on the wind-induced angle of attack and wind-shear velocity. A new method of reducing the state estimation filter dimension to yield suboptimal estimation of the three rigid-body states and two wind model states is applied and shown to cause only a slight additional increase in the standard deviations of the critical system variables.
TL;DR: Independent of the computational advantages, the iterative processing technique is useful for track management, permitting effective utilization of priority and interrupt schemes without disturbing the Kalman-filter operation.
Abstract: When the additive noise vector in the discrete observation process of a system can be partitioned into uncorrelated subvectors, an iterative processing technique for updating the Kalman-filter covariance matrix can often be used to increase computational efficiency. For standard typical programming algorithms and for a typical computer, the iterative processing technique can theoretically reduce the computational requirements of the covariance updating equation by over 50 percent. In practical situations, computational savings of over 30 percent are realizable, a significant amount particularly for real-time tracking applications in high-target-density environments. Furthermore, independent of the computational advantages, the iterative processing technique is useful for track management, permitting effective utilization of priority and interrupt schemes without disturbing the Kalman-filter operation.
TL;DR: The use of the extended Kalman filter as an approximate estimator for the states and parameters of nonlinear systems is well known and a decomposition is pointed out in this paper, which is possible with the usual augumentations of the state space by parameters.
Abstract: The use of the extended Kalman filter as an approximate estimator for the states and parameters of nonlinear systems is well known. A decomposition is pointed out in this letter, which is possible with the usual augumentations of the state space by parameters, and which may lead to a more efficient computing algorithm.
TL;DR: In this article, it was shown that there is one particular model yielding the smallest error-variance in a sense to be described, and that this model is causally invertible.
TL;DR: In this article, a coordination algorithm with one-step convergence for a number of subsystem Kalman estimators is proposed for sparsely coupled subsystems with few stochastic inputs.
Abstract: Sequential estimation of the states of several high-order interconnected systems may be prohibitive on computer time and storage if the problem is formulated as for a single system. Therefore, multilevel systems theory has been applied to derive a coordination algorithm, with one-step convergence, for a number of subsystem Kalman estimators. The procedure may be computationally attractive for sparsely coupled subsystems with few stochastic inputs.
TL;DR: In this article, a simple direct method of determining such a controller is presented, based on the fact that usually a linear combination of the set of state variables is all that is required to reconstruct the optimal control.
Abstract: Many optimal control solutions require a complete set of measurements of current state variables, which may not be fully available It is reasonable to ask whether compensators cannot be designed in such a way that the desirable qualities of the optimal control are reproduced One method of constructing a compensator that generates an asymptotically optimal control is to generate an estimate of the complete set of state variables by an auxiliary dynamic system, such as an observer or a Kalman filter It can be shown, however, that a simpler design is often possible by employing the fact that usually a linear combination of the set of state variables is all that is required to reconstruct the optimal control A simple direct method of determining such a controller is presented in this paper,
TL;DR: In this article, the extended Kalman-Bucy filtering algorithm is applied to the problem of estimating the system parameters of a single-channel missile attitude control system, in particular, noisy observations of body angular rate, normal acceleration, and control surface deflec tion are used in the algorithm to obtain estimates of BOR, angle of attack, control surface deformation, air density, and missile velocity.
Abstract: The extended Kalman-Bucy filtering algorithm is applied to the problem of estimating the system parameters of a single-channel missile attitude control system. In particular, noisy observations of body angular rate, normal acceleration, and control surface deflec tion are used in the algorithm to obtain estimates of body angular rate, angle of attack, control surface deflection, air density, and missile velocity. Computational results obtained for this problem are presented. These results are evaluated to determine the applicapility of this estimator as a part of an adaptive control scheme. In addition, a general approach to the formulation of practical nonlinear estimation problems is suggested.
TL;DR: In this paper, the structure of the upper atmosphere can be indirectly probed by light in order to determine the global density structure of ozone, aerosols, and neutral atmosphere, and the estimation of these densities is then performed using a linearized Kalman-Bucy filter and a linearised Kushner-Stratonovich filter.
Abstract: The structure of the upper atmosphere can be indirectly probed by light in order to determine the global density structure of ozone, aerosols, and neutral atmosphere. Scattered and directly transmitted light is measured by a satellite and is shown to be a nonlinear function of the state which is defined to be a point-wise decomposition of the density profiles. Dynamics are imposed on the state vector and a structured estimation problem is developed. The estimation of these densities is then performed using a linearized Kalman-Bucy filter and a linearized Kushner-Stratonovich filter.
TL;DR: Combined optimal estimation and control techniques are applied for the first time to satellite tracking systems of NASA, resulting in an estimated state of the satellite and of the tracking system.
Abstract: Combined optimal estimation and control techniques are applied for the first time to satellite tracking systems. Both radio antenna and optical tracking systems of NASA are considered. The optimal estimation is accomplished using an extended Kalman filter, resulting in an estimated state of the satellite and of the tracking system. This estimated state constitutes an input to the optimal controller. The optimal controller treats a linearized system with a quadratic performance index. The maximum principle is applied and a steady-state approximation to the resulting Riccati equation is obtained. A computer program, RATS, implementing this algorithm is described. A feasibility study of real-time implementation, tracking simulations, and parameter sensitivity studies are also reported.
TL;DR: The results of computer simulation of estimates using the Kalman filter are presented, as also on account of experiments with the actual reactor noise obtained on the Toshiba Training Reactor (TTR-1), using a nonlinear analog filter.
Abstract: Measurements of reactor parameters, such as prompt neutron generation time and subcriticality, have been made in the past using band pass filters or else applying the correlation technique. Other new methods of reactor parameter estimation would now appear to be available thanks to the recent developments in optimal filtering technique, such as optimal linear (Kalman) filter and some of the nonlinear optimal filters. Such new methods should prove very promising for on-line estimation of the reactor parameters. In this paper a description is given of the application of some of these new methods to subcriticality measurements. The results of computer simulation of estimates using the Kalman filter are presented, as also on account of experiments with the actual reactor noise obtained on the Toshiba Training Reactor (TTR-1), using a nonlinear analog filter. The results agreed with those reported by Nomura using conventional methods.
TL;DR: In this article, the authors consider the effect of modeling inaccuracies on optimal linear stochastic control systems and derive a covariance matrix composed of covariances of the estimates of the state variables, the errors in the estimates, and the correlation between these errors and the estimates.
Abstract: The deterioration of a linear optimal stochastic control scheme, designed under the assumptions of the certainty-equivalence principle (the optimal filter and controller, determined independently, combine to give a totally optimal system), is investigated when the parameters of the actual system do not coincide with the design values. This linear suboptimal stochastic system is described by a covariance matrix composed of covariances of the estimates of the state variables, the errors in the estimates of the state variables, and the correlation between these errors and the estimates. In particular, this paper is concerned with the covariance matrix resulting from a single state dynamical system and a scalar linear measurement function of both the state variable and the control variable (e.g., accelerometer measurements). A modeling error in the control variable coefficient of the measurement function may induce instability in the stochastic system with either unstable or stable dynamics. Furthermore, the absolute magnitude of the error in the control variable coefficient directly influences system stability, not the relative error. Thus, relatively small errors compared to the design value of this coefficient may be quite important. 4 LTHOUGH there are many studies on divergence of op-£^- tirnal filters, little attention has been given to the effect of modeling inaccuracies on optimal linear stochastic control systems. Here we extend Fitzgerald's1 investigation of Kalman filter divergence to optimal linear stochastic control systems. These systems are designed under the certainty equivalence principle2 which states that if the expected value of a quadratic function of the state and the control variables is to be minimized subject to linear dynamics, the optimal system is composed of an optimal filter in cascade with an optimal controller. This separation is possible because the estimate in the state is uncorrelated with the error in this estimate. If the parameters in the assumed model of the dynamics or the measurement device deviate from the parameters of the actual system, the estimate and the error in the estimate become correlated. The behavior of the system because of the gains based on an inaccurate model is studied by considering the coupled matrix covariance equation composed of the Covariances of the error in the estimate, the estimate, and the estimate with its error. Some of the characteristics of this linear matrix equation are studied through a scalar linear dynamic equation. The errors in system parameters enter into the 2X2 covariance equation in a dimensionless form allowing the following general results to be obtained: 1) The stochastic control system may be unstable when the nonoptimal filter and deterministic control systems individually are stable. 2) Instability occurs only when the error in the parameter exceeds a finite threshold value. 3) If the measurement is a linear function of the control variable as well as the state (e.g., accelerometer measurements) and there are errors in the coefficient of the control, then instability of the total system may occur for both stable and unstable dynamical systems. 4) The filter or control gains are not functions of the coefficient of the control variable in the measurement function. Consequently, Presented as Paper 70-36 at the AIAA 8th Aerospace Sciences
TL;DR: An approach to linear estimation through use of a "control" fed back into the system to cancel out the effect of disturbances or error signals is discussed, showing this approach to be suboptimal and is compared with the optimal with respect to estimation accuracy and sensitivity to modeling errors.
Abstract: This paper discusses an approach to linear estimation through use of a "control" fed back into the system to cancel out the effect of disturbances or error signals. Although this approach has very restricted application, it has found important usage in integrated navigation systems where one subsystem is an inertial measurement system. This approach is shown to be suboptimal and is compared with the optimal with respect to estimation accuracy and sensitivity to modeling errors. The feedback approach to estimation is shown to be similar to error estimation and correction in which the error states of the system are estimated and external correction applied. For discrete estimation using the feedback approach it is shown that error variance and Kalman gains for one-stage prediction should be used. Two examples are considered which compare the feedback approach to the optimum estimation approach. The system of the first example is quite simple, but provides simple analytical comparisons of the two estimation approaches. The second example system consists of a single-axis inertial guidance system and an independent position measuring system. Accuracy and sensitivity to modeling errors are compared. Other advantages and disadvantages of the two estimation approaches are discussed.
TL;DR: In this paper, a filter set is developed for estimating the state vector and observation error variances in a discrete-time linear system by use of empirical Bayes techniques, and the filter was found to converge fairly rapidly for the examples considered.
Abstract: A filter set is developed for estimating the state vector and observation error variances in a discrete-time linear system by use of empirical Bayes techniques. The error variances are assumed to be random and to vary over time. No initial conditions or distributional assumptions are required for the error variances, but all other assumptions for the Kalman filter are assumed to hold. The treatment is analytical, and a Monte Carlo simulation is used to verify the results. Graphs are presented which compare performance with the ideal case of known variances. The filter was found to converge fairly rapidly for the examples considered.
TL;DR: In this article, the authors extended the work of Nishimura to consider velocity-aided Kalman filtering for one-dimensional motion under random acceleration, and showed through examination of the steady state solution and the transient time constants that estimates incorporating velocity observations can be significantly improved over estimates based on range data alone.
Abstract: This paper extends recent work of Nishimura to consider velocity-aided Kalman filtering for one-dimensional motion under random acceleration. It is shown through examination of the steady-state solution and the transient time constants that estimates incorporating velocity observations can be significantly improved over estimates based on range data alone.
16 Aug 1971
30 Sep 1971
TL;DR: In this article, the maximum likelihood estimates of the unknown covariance parameters of a time-discrete nonstationary linear system are computed from measurement residuals of a suboptimal sequential filter.
Abstract: : The Kalman filter sequentially generates the minimum variance estimate of the state of a linear dynamic system. This estimate is a function of the covariance parameters of the dynamic system model which implies that these be known a priori. Unfortunately some or all these covariance parameters are often unknown in engineering applications of the Kalman filter. In the report the maximum-likelihood estimates of the unknown covariance parameters of a time-discrete nonstationary linear system are computed from measurement residuals of a suboptimal sequential filter. Results for nonstationary linear systems are useful for nonlinear systems because most nonlinear estimation problems are solved by linearization which results in linear nonstationary plant and measurement models.