Showing papers on "Kalman filter published in 1970"

On the identification of variances and adaptive Kalman filtering

Abstract: A Kalman filter requires an exact knowledge of the process noise covariance matrix Q and the measurement noise covariance matrix R . Here we consider the case in which the true values of Q and R are unknown. The system is assumed to be constant, and the random inputs are stationary. First, a correlation test is given which checks whether a particular Kalman filter is working optimally or not. If the filter is suboptimal, a technique is given to obtain asymptotically normal, unbiased, and consistent estimates of Q and R . This technique works only for the case in which the form of Q is known and the number of unknown elements in Q is less than n \times r where n is the dimension of the state vector and r is the dimension of the measurement vector. For other cases, the optimal steady-state gain K op is obtained directly by an iterative procedure without identifying Q . As a corollary, it is shown that the steady-state optimal Kalman filter gain K op depends only on n \times r linear functionals of Q . The results are first derived for discrete systems. They are then extended to continuous systems. A numerical example is given to show the usefulness of the approach.

Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets

Abstract: The majority of tactical weapons systems require that manned maneuverable vehicles, such as aircraft, ships, and submarines, be tracked accurately. An optimal Kalman filter has been derived for this purpose using a target model that is simple to implement and that represents closely the motions of maneuvering targets. Using this filter, parametric tracking accuracy data have been generated as a function of target maneuver characteristics, sensor observation noise, and data rate and that permits rapid a priori estimates of tracking performance to be made when maneuvering targets are to be tracked by sensors providing any combination of range, bearing, and elevation measurements.

Approaches to adaptive filtering

Abstract: In this expository paper, several approaches to Adaptive Filtering are discussed. The different methods are divided into four categories of (i) Bayesian Methods, (ii) Maximum Likelihood Methods, (iii) Correlation Methods, and (iv) Covariance-Matching Methods. The relationship between the methods and the difficulties associated with each method are described. New algorithms for the direct estimation of the optimal gain of a Kalman filter are given.

Optimal minimal-order observer-estimators for discrete linear time-varying systems

Abstract: This paper generalizes and unifies the concepts developed by Kalman and Luenberger pertaining to the design of discrete linear systems which estimate the state of a linear plant on the basis of both noise-free and noisy measurements of the output variables. Classes of minimal-order optimum "observer-estimators" are obtained which yield the conditional mean estimate of the state of the dynamical system. One explicit minimal-order optimal observer-estimator is constructed which generates one version of the conditional mean state estimate.

Theory and application of Kalman filtering

Abstract: As a unified extension of a group of related mathematical procedures, Kalman filtering is of assistance in the design of aircraft- and ground-based guidance and navigation data reduction and display systems.

Theory and applications of kalman filtering

Abstract: : Contents: Linear estimation theory; Further comments on the derivation of Kalman filters; Computational techniques in Kalman filtering; Modeling errors in Kalman filters; Suboptimal Kalman filter techniques; Comparison of Kalman, Bayesian and maximum likelihood estimation techniques; Nonlinear filtering and comparison with Kalman filtering; Linear smoothing techniques (post-flight data analysis); Nonlinear smoothing techniques; General questions on Kalman filtering in navigation systems; Application of Kalman filtering theory to augmented inertial navigation systems; Application of Kalman filtering to Baro/inertial height systems; Application of Kalman filtering to the C-5 guidance and control system; Application of Kalman filtering techniques to the Apollo program; Some applications of Kalman filtering in space guidance; Application of Kalman filtering for the alighnment of carrier aircraft inertial navigation systems; Navigation at sea using the invariants form of Kalman filtering; Marine applications of Kalman filtering; Optimal use of redundant information in an inertial navigation; Application of Kalman filtering techniques to strapdown system initia-alignment; and A Kalman filter augmented marine navigation system.

Nonlinear estimation with quantized measurements--PCM, predictive quantization, and data compression

Abstract: Statistics conditioned on quantized measurements are considered in the general case. These results are specialized to Gaussian parameters and then extended to discrete-time linear systems. The conditional mean of the system's state vector may be found by passing the conditional mean of the measurement history through the Kalman filter that would be used had the measurements been linear. Repetitive use of Bayes' rule is not required. Because the implementation of this result requires lengthy numerical quadrature, two approximations are considered: the first is a power-series expansion of the probablity-density function; the second is a discrete-time version of a previously proposed algorithm that assumes the conditional distribution is normal. Both algorithms may be used with any memory length on stationary or nonstationary data. The two algorithms are applied to the noiseless-channel versions of the PCM, predictive quantization, and predictive-comparison data compression systems; ensemble-average performance estimates of the nonlinear filters are derived. Simulation results show that the performance estimates are quite accurate for most of the cases tested.

A practical nondiverging filter

Abstract: A Kalmaii filter, and in fact all filters based on the usual approaches, have the troublesome problem that although they are convergent if the models and the statistics of the disturbances are correctly known, divergence can result when such knowledge is not available. This paper describes a method to remedy this problem by weighing past observations by a progressively small number. The result is a filter with an algorithm which differs from the Kalman filter in just one additional multiplication by a fixed scalar at each observation time. Examples document the improvements obtainable with the modified filter.

Application of Recursive Estimation and Kalman Filtering to Doppler Tracking

Abstract: The techniques of recursive estimation and Kalman filtering are applied to the problem of estimation of space vehicle orbits and trajectories using only measurements of the Doppler shifts in radio signals transmitted by or reflected from the target. Two approaches are described. One is a global fit technique in which the parameters of an assumed trajectory model are estimated. The second is a pointwise mapping technique in which the trajectory estimate is extrapolated as the target tracking data is received. The paper provides a tutorial illustration of the direct application of recursive optimization techniques to a class of engineering problems.

A survey of discrete Kalman-Bucy filtering with unknown noise covariances

Abstract: Discrete Kalman-Bucy linear filtering with inaccurate noise covariances, considering optimal and suboptimal systems, error analysis, probability theory, etc

Optimal stochastic control of nonlinear systems

Abstract: An algorithm is proposed for the feedback control of nonlinear systems, the observations of which are corrupted with noise of unknown statistics. The feedback loop contains a nonlinear Kalman filter, which produces sequential least-square estimates of the state of the system, and a controller designed to minimize an instantaneous performance criterion based on the state estimates. The scheme requires the on-line integration of n(n + 3)/2 differential equations, where n is the number of unknown states and parameters. The scheme is applied to the feedback control of a CSTR with a first-order exothermic reaction the temperature measurements of which are corrupted with random noise.

An Evaluation of a Pilot Model Based on Kalman Filtering and Optimal Control

Abstract: A pilot model based on Kalman filtering and optimal control is given which, because of its structure, provides for estimation of the plant state variables, the forcing functions, the time delay, and the neuromuscular lag. The inverse filter and control problem is considered where the noise and cost function parameters yield a frequency response which is in close agreement with that found experimentally. A good correspondence with sine-wave tracking is shown including "eyes closed" tracking.

Sequential structure and parameter-adaptive pattern recognition--I: Supervised learning

Abstract: Bayes optimal sequential structure and parameter-adaptive pattern-recognition systems for continuous data are derived. Both off-line (or prior to actual operation) and on-line (while in operation) supervised learning is considered. The concept of structure adaptation is introduced and both structure as well as parameter-adaptive optimal pattern-recognition systems are obtained. Specifically, for the class of supervised-learning pattern-recognition problems with Gaussian process models and linear dynamics, the adaptive pattern-recognition systems are shown to be decomposable ("partition theorem") into a linear nonadaptive part consisting of recursive matched Kalman filters, a nonlinear part--a set of probability computers--that incorporates the adaptive nature of the system, and finally a part of the correlator-estimator (Kailath) form. Extensions of the above results to the M -ary hypotheses case where M \geq 2 are given.

Optimal simultaneous state estimation and parameter identification in linear discrete-time systems

Abstract: The topic of this paper is the simultaneous estimation of state and parameters in linear discrete-time dynamic systems. The system is subject to a known arbitrary input (control), a random input (additive driving noise) and the output observation is also contaminated by noise. The noises are Gaussian, zero-mean, independent and with known variances. The problem is formulated under the assumption that the system parameters are unknown constants. Previous works in the literature treated this problem assuming that each parameter can take values over a finite set with known a priori probabilities. The proposed scheme yields the maximum a posteriori and maximum likelihood estimates for the system's state and parameters, respectively. They are obtained by solving the likelihood equations-a system of nonlinear equations with the state and parameters as unknowns. Use is made of the fact that the dynamical system considered is linear and the problem is separated into two interconnected linear problems: one for the state, the other for the parameters. The solution is obtained by iterating between two systems of linear equations. The estimation technique presented is optimal in the following sense: no approximations are involved and the estimates of the parameters converge to the true values at the fastest possible rate, as given by the Cram�r-Rao lower bound, i.e. they are asympotically efficient. This is proved, using a theorem which states that, under certain general conditions, the maximum likelihood estimate with dependent observations is consistent and asymptotically efficient. The problem of uniqueness of the solution is discussed for the case of a scalar unknown parameter. Use is made of a theorem due to Perlman, generalized for the case of dependent observations. Due to the fact that the estimation-identification is done in the presence of input and output noise and an arbitrary known input, the procedure can be considered an on-line technique. Since estimates are available after each measurement, this estimationidentification procedure is suited for use in the adaptive control of unknown (or partially known) linear plants.

Transient Response of Tracking Filters with Randomly Interrupted Data

Abstract: The transient responses during the initialization phase of a first-order ?-s tracking filter and a second-order Kalman filter are evaluated as a function of radar measurement accuracy and the probability of receiving valid data at the prescribed intervals. Monte Carlo simulation results are complemented by analysis of the filtering processes and curves are presented which clearly define the deterioration in filter performance attributable to reduced probabilities of data acquisition. In addition, the responses of ?-s and Kalman filters are shown to be identical when the ?, s gains are selected optimally.

Sensitivity analysis of discrete Kalman filters.

Abstract: This paper investigates the sensitivity of discrete Kalman filters to erroneous models. Both parameter and structure (state dimensionality) sensitivity are considered, as well as deterministic and random parameter errors. Iterative algorithms are derived for the calculation of the actual filter error covariance matrix for the case of known (deterministic) modelling errors. For the case of random statistical and dynamical modelling errors, an optimal mean-square error estimate of the actual system performance is derived.

Performance measure for adaptive Kalman estimators

Abstract: An on-line performance measure for adaptive Kalman estimators is derived which is shown to be an optimal mean-square estimate of actual system performance. The proposed measure utilizes quantities readily available from the adaptive estimator, and hence a minimum of additional computational effort is required for its evaluation.

Modeling errors in Kalman filters

Abstract: Algorithms for error effects due to modeling errors in Kalman filter for continuous and discrete systems

Data Assimilation Schemes For Non-linear Shallow Water Flow Models

Abstract: In theory K aim an filters can be used to solve many on-line data assimilation problems. However, for models resulting from the discretization of partial differential equations the number of state variables is usually very large, leading to a huge computational burden. Therefore approximation of the Kalman filter equations is in general necessary. In this paper two new algorithms are proposed, that extend the idea of the Reduced Rank Square Root filter [15] for use with non-linear models. The algorithms are based on a low rank approximation of the error covariance matrix and use a square root representation of the error covariance. For both algorithms the tangent linear model is not needed. The first algorithm proposed is accurate up to first order terms, which is comparable to the extended Kalman filter. The second, at the cost of twice the number of computations, is second order accurate, which may be important for strongly nonlinear models. Several experiments were performed on a model of the southern part of the North Sea to measure the performance of both algorithms. Both algorithms perform well when the the number of modes, i.e. the rank of the approximation, is set to 30. This corresponds to a computation time of approximately 30 model runs for the first order algorithm and 60 for the second order algorithm.

Relationship Between State-space And Input-outputModels Via Observer Markov Parameters

Abstract: This paper describes the relationship between two types of commonly used models in control and identification theory: state-space and input-output models. The relationship between the two model structures can be explained in terms of a newly formulated set of parameters called the observer Markov parameters. This is different from the usual connection between the two model structures via well-known canonical realizations. The newly defined observer Markov parameters generalize the standard system Markov parameters by incorporating information of an associated observer. In the deterministic case, the observer Markov parameters subsume a state-space model and a deadbeat observer gain In the stochastic case, the observer Markov parameters contain information of an optimal observer such as a Kalman filter. The observer Markov parameters can be identified directly from experimental input-output data. Making use of the new connection, one can extract from the identified observer Markov parameters not only a state-space model of the system, but also an associated observer gain for use in modern state-space based feedback controller design.

Nonlinear filtering techniques with application to strapdown computation

Abstract: Applying nonlinear filtering techniques to the attitude computation in a strapdown inertial navigation system is discussed. A special class of nonlinear filtering problems is formulated to accommodate the physical problem involving the attitude computation. Recursive filtering algorithms which take advantage of the special characteristics of this mathematical model are derived. Numerical works based on the Monte Carlo simulation are given and comparisons are made with a commonly used algorithm.

Remote computer control of an industrial process

Abstract: The development of a nonlinear dynamic model of an industrial process is summarized. The model includes additive stochastic terms and not all the state variables were accessible. Nonlinear state estimation was approximated by a linearized Kalman filter and the control algorithm was from dynamic programming. The development of software and hardware for a remote on-line computer control experiment is then described, in which the industrial plant was connected to a process control computer some 130 miles away by a regular telephone channel.

On the optimal control of linear systems with incomplete information

Abstract: : The control of linear systems with incomplete information is considered where the unknown disturbances and/or random parameters are assumed to satisfy some statistical laws. The observer theory for linear systems is developed which generalizes the concepts due to Kalman and Luenberger pertaining to the design of linear systems which estimate the state of a linear plant on the basis of both noise-free and noisy measurements of the output variables. The separation theorem for linear system is then extended for such observers- estimators. The problem of controlling a linear system with unknown gain is then considered. An open-loop-feedback-optimal control algorithm is developed which seems to be computationally feasible. Existence of such suboptimal control scheme is proved under the assumption that the uncertainties in the unknown gail will not grow in time. Convergence of such suboptimal control system to the truly optimal control system is considered. A computer program is developed to study the control of a variety of third order systems with known poles but unknown zeroes. The experimental results serve to provide more insights into the structure and behavior of the open-loop-feedback-optimal control systems.

Suboptimization of a Kalman Filter with Delayed-States as Observables

Abstract: : The computational requirements of the Kalman filter may become excessive when the measurement model includes a connection with both the present state and a previous state. Three aspects of this problem are studied. A performance analysis is presented and a performance index is defined as an aid in evaluating the performance of two classes of suboptimal filters which may be used to solve this problem. The two classes are suboptimality due to modeling variations and due to alternate gain algorithms. A suboptimal filter is derived which belongs to the second class. The simulation of a proposed integrated inertial/doppler-satellite navigation system is performed to study the performance of filters belonging to both of the above classes.

Discrete-Time Fixed-Lag Smoothing,

Abstract: : It has been shown that previously derived algorithms for recursive, optimal, fixed-lag smoothing are unstable under the same assumptions for which the corresponding Kalman filter is stable. In this paper, a stable algorithm for fixed-lag smoothing is developed for the discrete-time problem. The algorithm is optimal and appears to be computationally tractable for on-line smoothing. A smoothing error covariance relation is also developed in the paper to provide a means for assessing fixed-lag smoothing performance. A numerical example involving a third-order communications system model is presented and the performance of fixed-lag smoothing is compared with that of filtering for two separate cases. (Author)