Showing papers on "Invariant extended Kalman filter published in 1991"

Identification of observer/Kalman filter Markov parameters: Theory and experiments

Abstract: An algorithm to compute Markov parameters of an observer or Kalman filter from experimental input and output data is discussed The Markov parameters can then be used for identification of a state space representation, with associated Kalman gain or observer gain, for the purpose of controller design The algorithm is a non-recursive matrix version of two recursive algorithms developed in previous works for different purposes The relationship between these other algorithms is developed The new matrix formulation here gives insight into the existence and uniqueness of solutions of certain equations and gives bounds on the proper choice of observer order It is shown that if one uses data containing noise, and seeks the fastest possible deterministic observer, the deadbeat observer, one instead obtains the Kalman filter, which is the fastest possible observer in the stochastic environment Results are demonstrated in numerical studies and in experiments on an ten-bay truss structure

Design and implementation of an extended Kalman filter for the state estimation of a permanent magnet synchronous motor

Abstract: Practical considerations for implementing the discrete extended Kalman filter in real time with a digital signal processor are discussed. The system considered is a permanent magnet synchronous motor (PMSM) without a position sensor, and the extended Kalman filter is designed for the online estimation of the speed and rotor position by only using measurements of the motor voltages and currents. The algorithms developed to allow efficient computation of the filter are presented. The computational techniques used to simplify the filter equations and their implementation in fixed-point arithmetic are discussed. Simulation and experimental results are presented to demonstrate the feasibility of this estimation process. >

Observers for induction motor state and parameter estimation

Abstract: The Kalman filter in its basic form is a state estimator and can be applied to the problem of estimating induction motor rotor currents in a vector control scheme. This filter is shown to combine information from the plant model with output measurements to produce an optimal estimate of the unmeasured states. Also described is the application of the extended Kalman filter algorithm to the online estimation of rotor resistance in an induction motor drive. Significant savings in computing requirements are obtained with a reduced-order model of the motor, in which measured, rather than computed, values of stator currents are used. >

Simplification of the Kalman filter for meteorological data assimilation

Abstract: The paper proposes a new statistical method of data assimilation that is based on a simplification of the Kalman filter equations. The forecast error covariance evolution is approximated simply by advecting the mass-error covariance field, deriving the remaining covariances geostrophically, and accounting for external model-error forcing only at the end of each forecast cycle. This greatly reduces the cost of computation of the forecast error covariance. In simulations with a linear, one-dimensional shallow-water model and data generated artificially, the performance of the simplified filter is compared with that of the Kalman filter and the optimal interpolation (OI) method. The simplified filter produces analyses that are nearly optimal, and represents a significant improvement over OI.

The Behavior of Forecast Error Covariances for a Kalman Filter in Two Dimensions

Abstract: The behavior of forecast error covariances in a fairly realistic setting is demonstrated via a Kalman filter algorithm. It is used to assimilate simulated data from the existing radiosonde network, from the demonstration network of 31 Doppler wind profilers in the central U.S., and from hypothetical radiometers located at five of the profiler sites. Some theoretical justification of the hypothesis advanced by Phillips (1982), and the hypothesis is used to formulate the model error covariance matrix required by the Kalman filter. The results show that assimilating the profiler wind data leads to a large reduction of forecast/analysis error in heights as well as in winds, over the profiler region and also downstream, when compared with the results of assimilating the radiosonde data alone. The forecast error covariance matrices that the Kalman filter calculates to obtain this error reduction differ considerably from those prescribed by the optimal interpolation schemes that are employed for data assimilation at operational centers.

Efficiency of the extended Kalman filter for nonlinear systems with small noise

Abstract: The problem of nonlinear filtering is studied asymptotically as the noise tends to zero Detectability conditions ensuring that the filtering error tends to zero are derived, and under these conditions, it is proved that the extended Kalman filter provides a good suboptimal filter; the smoothing problem is also studied The proofs use some estimates for the linearization of stochastic systems, some changes of probability, and stochastic differentiation techniques

An alternative derivation of the modified gain function of Song and Speyer

Abstract: A simpler derivation of the modified gain function for the bearings-only measurement problem as defined by T.L. Song et al. (see ibid vol.AC-30, no.30 (1985)) is presented. The form which results show the relationship between the modified gain and the standard gain. The relationship between the modified gain and the standard gain of an extended Kalman filter is shown. It is confirmed that the modified gain extended Kalman filter performs better than the standard extended Kalman filter. >

Initializing the kalman filter for nonstationary time series models

Abstract: . The problem of initializing the Kalman filter for nonstationary time series models is considered. Ansley and Kohn have developed a ‘modified Kalman filter’ for use with nonstationary models to produce estimates from what they call a ‘transformation approach’. We show that the same results can be obtained with a suitable initialization of the ordinary Kalman filter. Assuming there are d starting values for the nonstationary series, we initialize the Kalman filter using data through time d with the transformation approach estimate of the state vector and its associated error covariance matrix at time d. We give details of the initialization for ARIMA models, ARIMA component models and dynamic linear models. We present an example to illustrate how the results may differ from results obtained under more naive initializations that have been suggested.

Streamlining measurement iteration for EKF target tracking

Abstract: The estimation problem is defined, and a review of how the linear estimation approach of Kalman filtering is extrapolated to form an extended Kalman filter (EKF), applicable for state estimation in nonlinear systems is presented. A mechanization of an EKF variation known as an iterated EKF, offering improved tracking performance, is treated. A streamlined version of an iterated EKF that has a lesser computational burden (fewer operations per cycle or time step) than prior formulations is offered. A nonlinear filtering application example, to be used as a testbed for this new approach, is described, and the detailed modeling considerations as needed for exoatmospheric random-variable radar target tracking are discussed. The performance of the streamlined mechanization is illustrated in this radar target tracking example, and comparisons are made with the performance of an EKF without measurement iteration. >

A Kalman filter approach for accurate 3-D motion estimation from a sequence of stereo images

Abstract: This paper presents a Kalman filter approach for accurately estimating the 3-D position and orientation of a moving object from a sequence of stereo images. Emphasis is given to finding a solution for the following problem incurred by the use of a long sequence of images: the images taken from a longer distance suffer from a larger noise-to-signal ratio, which results in larger errors in 3-D reconstruction and, thereby, causes a serious degradation in motion estimation. To this end, we have derived a new set of discrete Kalman filter equations for motion estimation: (1) The measurement equation is obtained by analyzing the effect of white Gaussian noise in 2-D images on 3-D positional errors (instead of directly assigning Gaussian noise to 3-D feature points) and by incorporating an optimal 3-D reconstruction under the constraints of consistency satisfaction among 3-D feature points. (2) The state propagation equation, or the system dynamic equation, is formulated by describing the rotation between two consecutive 3-D object poses, based on quaternions and representing the error between the true rotation and the nominal rotation (obtained by 3-D reconstruction) in terms of the measurement noise in 2-D images. Furthermore, we can estimate object position from the estimation of object orientation in such a way that an object position can be directly computed once the estimation of an object orientation is obtained. Simulation results indicate that the Kalman filter equations derived in this paper represent an accurate model for 3-D motion estimation in spite of the first-order approximation used in the derivation. The accuracy of this model is demonstrated by the significant error reduction in the presence of large triangulation errors in a long sequence of images and by a shorter transition period for convergence to the true values.

Adaptive Kalman filtering

Abstract: The increased power of small computers makes the use of parameter estimation methods attractive. Such methods have a number of uses in analytical chemistry. When valid models are available, many methods work well, but when models used in the estimation are in error, most methods fail. Methods based on the Kalman filter, a linear recursive estimator, may be modified to perform parameter estimation with erroneous models. Modifications to the filter involve allowing the filter to adapt the measurement model to theexperimental data through matching the theoretical and observed covoriance of the filter innovations sequence. The adaptive filtering methods that result have a number of applications in analytical chemistry.

A two-stage Kalman estimator for state estimation in the presence of random bias and for tracking maneuvering targets

Abstract: The authors provide the optimal solution of a two-stage estimation problem in the presence of random bias. Under an algebraic constraint, the optimal estimate of the system state can be obtained as a linear combination of the output of the first stage (a bias-free filter) and the second stage (a bias filter). The results presented provide a basis for assessing the suboptimality of a two-stage estimator when used for a specific system. By treating the bias vector as a target acceleration, the two-state Kalman estimator can be used for tracking maneuvering targets. >

Data processing system using a kalman filter

Abstract: For processing data relating to the performance of an apparatus, the data is analyzed using a Kalman Filter. After a first pass of data through the filter, the results are refined by discarding at least one less significant component performance change and/or sensor bias. The Kalman Filter is then re-run using the modified data. As further runs of the Kalman Filter are performed, as required, the input of each successive run is refined by discarding from the preceding run at least one further component performance change and/or sensor bias. For each run, an objective function is evaluated for the amount of unexplained measurement change and/or the amount of component performance change and sensor bias. The run whose results show an acceptable value for the objective function is selected as the best solution. In this way, the tendency of the Kalman Filter to distribute the cause of any sensed performance change over all the possible sources of that change is avoided. The sets of measurement data are then analyzed to determine levels and/or trends in component performance and sensor bias.

Square root parallel Kalman filtering using reduced-order local filters

Abstract: The basic parallel Kalman filtering algorithms derived by H.R. Hashemipour et al. (IEEE Trans. Autom. Control. vol.33, p.88-94, 1988) are summarized and generalized to the case of reduced-order local filters. Measurement-update and time-update equations are provided for four implementations: the conventional covariance filter, the conventional information filter, the square-foot covariance filter, and the square-foot information filter. A special feature of the suggested architecture is the ability to accommodate parallel local filters that have a smaller state dimension than the global filter. The estimates and covariance or information matrices (or their square roots) from these reduced-order filters are collated at a central filter at each step to generate the full-size, globally optimal estimates and their associated error covariance or information matrices (or their square roots). Aspects of computational complexity and the ensuing tradeoff with communication are discussed. >

Bearings-only tracking: a hybrid coordinate system approach

Abstract: A bearings-only tracking algorithm is described. The algorithm is an extended Kalman filter (EKF) which combines the linear-in-state properties of the Cartesian state variable definition with the linear-in-measurement properties of the modified polar (MP) state variable definition. This hybrid approach uses the Cartesian system for state and state covariance extrapolation and uses the MP system for state and state covariance updating. Accurate state and state covariance extrapolation is achieved without numerical integration. The filter equations of this EKF are, furthermore, derived using a line-of-sight algebra which yields equations which are nonlinear algebraic rather than transcendental. This approach allows for easier of the filter equations and provides for greater insight into the bearings-only tracking problem. Applying recent results from differential geometric system theory, observability is analyzed by applying the Lie bracket criteria to the algebraic filter equations. >

Predictability and unpredictability in Kalman filtering

Abstract: The authors study the dynamical behavior of the Kalman filter when the given parameters are allowed to vary in a way which does not necessarily correspond to an underlying stochastic system. This may correspond to situations in which the basic parameters are chosen incorrectly through estimates. The authors show that, as has been suggested by Kalman, the filter equations converge to a limit (corresponding to a steady-state filter) for a subset of the parameter space which is much larger than that corresponding to bona fide stochastic systems. More surprisingly, in the complement of this subset, the filtering equations behave in both a regular and an unpredictable manner, representative of some of the basic aspects of chaotic dynamics. This interesting dynamical behavior occurs already for one-dimensional filters, and a complete phase portrait in this case is given. >

Maneuvering target tracking using extended Kalman filter

Abstract: A numerically well-conditioned, quasi-extended Kalman filter is proposed. The filter is numerically described. The simulation results presented show that the estimation performance of the quasi-extended filter is superior, for short distances, compared with the widely used linear tracking filters. In addition, the simplicity of the quasi-extended filter makes it very easy to implement. >

Linear Systems Over Rings: From R. E. Kalman to the Present

Abstract: The paper begins with a brief history of the role played by R.E. Kalman in the establishment of the field of linear systems over a commutative ring. The theory of systems over rings is motivated by considering integer systems, systems with time delays, parameter-dependent systems, and multidimensional systems including spatially-distributed systems. Then a brief survey is given of existing work on the control of systems over rings along with some discussion of open research problems.

Kalman filter mechanization for INS airstart

Abstract: A strapdown mechanization and associated Kalman filter were developed to provide both ground align and airstart capabilities for inertial navigation systems using Doppler velocity and position fixes, while not requiring an initial heading estimate. This is accomplished by use of a Doppler velocity sensor and a position source such as the global positioning system (GPS) manual fly-over update or target sighting systems. Filter transition from coarse to fine align mode is accomplished without disrupting the estimation of the inertial instrument or aiding sensor errors, by defining the azimuth error state as wander angle error, and using a simple, but effective, manipulation of the filter covariance matrix. >

Extended Kalman Filter and System Identification

Abstract: The Kalman filtering process has been designed to estimate the state vector in a linear model. If the model turns out to be nonlinear, a linearization procedure is usually performed in deriving the filtering equations. We will consider a real-time linear Taylor approximation of the system function at the previous state estimate and that of the observation function at the corresponding predicted position. The Kalman filter so obtained will be called the extended Kalman filter. This idea to handle a nonlinear model is quite natural, and the filtering procedure is fairly simple and efficient. Furthermore, it has found many important real-time applications. One such application is adaptive system identification which we will also discuss briefly in this chapter. Finally, by improving the linearization procedure of the extended Kalman filtering algorithm, we will introduce a modified extended Kalman filtering scheme which has a parallel computational structure. We then give two numerical examples to demonstrate the advantage of the modified Kalman filter over the standard one in both state estimation and system parameter identification.

Parallel and sequential block Kalman filtering and their implementations using systolic arrays

Abstract: Two sets of block Kalman filtering equations that differ in the manner of generating the initial and updated estimates are derived. Parallel and sequential schemes for generating these estimates are adopted. It is shown that the parallel implementation inherently leads to a block Kalman estimator which provides filtered estimates at the vector (block) level and fixed-lag smoother estimates at the sample level. The sequential implementation scheme, on the other hand, generates the estimates of each sample recursively, leading naturally to a scalar (filter) estimator. These scalar estimates are arranged in a vector form, resulting in a block estimator which solely generates filtered estimates both at the vector and sample levels. Simulation results on a speech signal are presented which indicate the advantages of the sequential block Kalman filter. An algorithm for iterative calculation of Kalman gain and error covariance matrices is given which does not require any matrix inversion operation. The implementation of this algorithm using available systolic array processors is presented. A ring systolic array which can be used to implement the state update part of the block Kalman filter is suggested. >

A synopsis of the smoothing formulae associated with the Kalman Filter

Abstract: This paper provides straightforward derivations of a wide variety of smoothing formulae which are associated with the Kalman filter. The smoothing operations are of perennial interest in the fields of communications engineering and signal processing. Recently they have begun to interest statisticians and economists. It is often asserted that it is tedious and difficult to derive the formulae. We show that this need not be so. Citation Copyright 1993 by Kluwer Academic Publishers. (This abstract was borrowed from another version of this item.)

On the optimal design of the output transformation for discrete-time linear systems

Abstract: The optimization of the output matrix for a discrete-time, single-output, linear stochastic system is approached from two different points of view. Firstly, we investigate the problem of minimizing the steady-state filter error variance with respect to a time-invariant output matrix subject to a norm constraint. Secondly, we propose a filter algorithm in which the output matrix at timek is chosen so as to maximize the difference at timek+1 between the variance of the prediction error and that of the a posteriori error. For this filter, boundedness of the covariance and asymptotic stability are investigated. Several numerical experiments are reported: they give information about the limiting behavior of the sequence of output matrices generated by the algorithm and the corresponding error covariance. They also enable us to make a comparison with the results obtained by solving the former problem.

Speech enhancement based on Kalman filtering and EM algorithm

Abstract: Speech enhancement via Kalman filtering is considered. It is generally agreed that the quality of the estimate of speech production model parameters is crucial to the performance of the Kalman filter. The Kalman filter with a more accurate estimate of the LPC parameters will generally achieve better noise cancellation results. In practice only the noisy speech is available for the LPC analysis. Then the estimate of the LPC parameters is usually inaccurate, which in turn degrades the performance of the Kalman filter. In order to overcome the problem, a Kalman filtering scheme applied in conjunction with the EM algorithm is proposed. Simulation results demonstrate the expected performance improvement in terms of signal-to-noise ratio (SNR) gains by the new method. >

Lp‐stability of estimation errors of kalman filter for tracking time‐varying parameters

Abstract: One presents a detailed analysis for the L p -stability of tracking errors when the Kalman filter is used for tracking undknown time-varying parameters. The results of this paper differ from the previous ones in that the regression vector (in a linear regression model) or the output matrix (in state space terminology) is random rather than deterministic. The context is kept general so that, in particular, the time-varying parameter is allowed to be unbounded, and no assumption of stationarity or independence for signals is made