Showing papers on "Invariant extended Kalman filter published in 1994"

Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics

Abstract: A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computational load for reasonable accuracy is only a fraction of what is required for the extended Kalman filter and is given by the storage of, say, 100 model states for an ensemble size of 100 and thus CPU requirements of the order of the cost of 100 model integrations. The proposed method can therefore be used with realistic nonlinear ocean models on large domains on existing computers, and it is also well suited for parallel computers and clusters of workstations where each processor integrates a few members of the ensemble.

Robust Kalman filtering for uncertain discrete-time systems

Abstract: This paper is concerned with the problem of a Kalman filter design for uncertain discrete-time systems. The system under consideration is subjected to time-varying norm-bounded parameter uncertainty in both the state and output matrices. The problem addressed is the design of a linear filter such that the variance of the filtering error is guaranteed to be within a certain bound for all admissible uncertainties. Furthermore, the guaranteed cost can be optimized by appropriately searching a scaling design parameter. >

Advanced data assimilation in strongly nonlinear dynamical systems

Abstract: Advanced data assimilation methods are applied to simple but highly nonlinear problems. The dynamical systems studied here are the stochastically forced double well and the Lorenz model. In both systems, linear approximation of the dynamics about the critical points near which regime transitions occur is not always sufficient to track their occurrence or nonoccurrence. Straightforward application of the extended Kalman filter yields mixed results. The ability of the extended Kalman filter to track transitions of the double-well system from one stable critical point to the other depends on the frequency and accuracy of the observations relative to the mean-square amplitude of the stochastic forcing. The ability of the filter to track the chaotic trajectories of the Lorenz model is limited to short times, as is the ability of strong-constraint variational methods. Examples are given to illustrate the difficulties involved, and qualitative explanations for these difficulties are provided. Three generalizations of the extended Kalman filter are described. The first is based on inspection of the innovation sequence, that is, the successive differences between observations and forecasts; it works very well for the double-well problem. The second, an extension to fourth-order moments, yields excellent results for the Lorenz model but will be unwieldy when applied to models with high-dimensional state spaces. A third, more practical method--based on an empirical statistical model derived from a Monte Carlo simulation--is formulated, and shown to work very well. Weak-constraint methods can be made to perform satisfactorily in the context of these simple models, but such methods do not seem to generalize easily to practical models of the atmosphere and ocean. In particular, it is shown that the equations derived in the weak variational formulation are difficult to solve conveniently for large systems.

Multiscale systems, Kalman filters, and Riccati equations

Abstract: An algorithm analogous to the Rauch-Tung-Striebel algorithm/spl minus/consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step/spl minus/was developed previously by the authors (ibid. vol.39, no.3, p.464-78 (1994)). In this paper they present a detailed system-theoretic analysis of this filter and of the new scale-recursive Riccati equation associated with it. While this analysis is similar in spirit to that for standard Kalman filters, the structure of the dyadic tree leads to several significant differences. In particular, the structure of the Kalman filter error dynamics leads to the formulation of an ML version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the Hamiltonian for the smoothing problem. In addition, the notion of stability for dynamics requires some care as do the concepts of reachability and observability. Using these system-theoretic constructs, the stability and steady-state behavior of the fine-to-coarse Kalman filter and its Riccati equation are analysed. >

A 3D State Space Formulation of a Navigation Kalman Filter for Autonomous Vehicles

Abstract: : The Kalman Filter has many applications in mobile robotics ranging from perception, to position estimation, to control. This report formulates a navigation Kalman Filter. That is, one which estimates the position of autonomous vehicles. The filter is developed according-to the state space formulation of Kalman's original papers. The state space formulation is particularly appropriate for the problem of vehicle position estimation. This filter formulation is fairly general. This generality is possible because the problem has been addressed

Identification and Uncertainty Estimation of Structural Parameters

Abstract: In the application of system identification to a structural system, unknown parameters are determined based on the numerical analysis of input and output measurements. The accuracy of an identified parameter and its uncertainty both depend on the numerical method, measurement noise and modeling error. Most studies, however, identify parameter means without addressing the issue of parameter uncertainties. Presented in this paper is an improved version of the commonly used extended Kalman filter (EKF) by incorporating an adaptive filter procedure. The system noise covariance is updated in time segments in order to ensure statistical consistency between the predicted error covariance and the mean square of actual residuals. Comprising two stages in a cycle, the adaptive EKF method not only identifies the parameter values but also gives a useful estimate of uncertainties. Two numerical examples of simulation with noise are presented. The first example illustrates the superior statistical performance of the pr...

Interlaced Kalman filtering of 3D angular motion based on Euler's nonlinear equations

Abstract: A novel Kalman filtering technique is presented that reduces the mean-square-error (MSE) between three-dimensional (3D) actual angular velocity values and estimated ones by an order of magnitude (when compared with the MSE resulting from direct measurements) even under extremely low signal-to-noise ratio conditions. The filtering problem is nonlinear in nature because the dynamics of 3D angular motion are described by Euler's equations. This nonlinear set of differential equations state that the angular acceleration in one axis is proportional to the torque applied to that axis, and to the products of angular velocity components in the other two axes of rotation. Instead of using extended Kalman filtering techniques to solve this complex problem, the authors developed a new approach where the nonlinear Euler's model is decomposed into two pseudolinear models (primary and secondary). The first model describes the time progression of the state vector containing the linear terms, while the other characterizes the propagation of the state vector containing the nonlinearities. This makes it possible to run two interlaced discrete-linear Kalman filters simultaneously. One filter estimates the values of the state vector containing the linear terms, while the other estimates the values of the state vector containing the nonlinear terms in the system. These estimates are then recombined, solving the nonlinear estimation process without linearizing the system. Thus, the new approach takes advantage of the simplicity, computational efficiency and higher convergence speed of the linear Kalman filter form and it overcomes many of the drawbacks typical of conventional extended Kalman filtering techniques. The high performance and effectiveness of this method is demonstrated through a computer simulation case study. >

Incremental least squares methods and the extended Kalman filter

Abstract: Proposes and analyzes nonlinear least squares methods, which process the data incrementally, one data block at a time. Such methods are well suited for large data sets and real time operation, and have received much attention in the context of neural network training problems. The author focuses on the extended Kalman filter, which may be viewed as an incremental version of the Gauss-Newton method. The author provides a nonstochastic analysis of its convergence properties, and discusses variants aimed at accelerating its convergence. >

Application of neural networks in target tracking data fusion

Abstract: Kalman filtering is a fundamental building block of most multiple-target tracking (MTT) algorithms. The other building block usually involves some type of data association schemes. Here it is proposed to incorporate a neural network into the normal Kalman filter configuration such that the neural network provides the adaptive capability the filter needs. As such the estimation error of the Kalman filter would be reduced, hence improving the MTT solution. Simulation results have shown that this claim is valid. >

Training controllers for robustness: multi-stream DEKF

Abstract: Kalman-filter-based training has been shown to be advantageous in many training applications. By its nature, extended Kalman filter (EKF) training is realized with instance-by-instance updates, rather than by performing updates at the end of a batch of training instances or patterns. Motivated originally by the desire to be able to base an update an a collection of instances, rather than just one, we recognized that the simple construct of multiple streams of training examples allows a batch-like update to be performed without violating an underlying principle of Kalman training, vis. that the approximate error covariance matrix remain consistent with the updates that have actually been performed. In this paper, we present this construct and show how it may be used to train robust controllers, i.e. controllers that perform well for a range of plants. >

Design of an extended Kalman filter frequency tracker

Abstract: This paper discusses the problem of designing a frequency tracker using the extended Kalman filter (EKF). The design trade-off between balancing noise rejection and tracking at a maximal slew rate is considered, as are the effect of varying design noise covariance values. The performance penalties for over- and under-design of noise covariances are examined and theoretically supported design guidelines are suggested. These are illustrated via simulation results. >

An EKF-based joint estimator for interference, multipath, and code delay in a DS spread-spectrum receiver

Abstract: In direct-sequence (DS) spread-spectrum communications, it is often necessary to obtain pseudo-noise (PN) code synchronization in the presence of both narrowband interference and frequency-selective multipath. However, conventional delay-lock code tracking loops are not optimized for such applications, nor are they well-suited to digital implementations. A code tracking algorithm based on the extended Kalman filter (EKF) is described that provides both code synchronization and joint estimates of interferer and channel parameters. It is first assumed that the interference can be modeled by an N-th order autoregressive (AR) process, and the multipath by a finite impulse response filter. A composite channel, equivalent to the convolution of the prewhitening filter and multipath coefficients is then constructed. The received waveform is shown to be a linear function of the composite channel parameters, that can therefore be directly estimated by an extended Kalman filter. The code delay /spl tau/ is viewed as a nonlinear parameter, that can likewise by estimated, after an appropriate linearization, using the EKF. The performance of the algorithm is first evaluated by computing the average bit-error rate (BER) of a digital RAKE receiver, that employs the joint delay, channel and interferer estimates derived from the EKF. In addition, a nonlinear analysis of the EKF is performed in which the code delay error and error covariance are modeled as a two-dimensional Markov process. The joint probability density function of these parameters, and nonlinear delay error variance, are obtained via numerical integration of a two-dimensional Chapman-Kolmogorov equation. >

On the Nonlinear Dynamics of Fast Filtering Algorithms

Abstract: The main purpose of this paper is to address a fundamental open problem in linear filtering and estimation, namely, what is the steady-state or asymptotic behavior of the Kalman filter, or the Kalman gain, when the observed stationary stochastic process is not generated by a finite-dimensional stochastic system, or when it is generated by a stochastic system having higher-dimensional unmodeled dynamics. For example, some time ago Kalman pointed out that the usual positivity conditions assumed in the classical situation are not in fact necessary for the Kalman filter to converge. Using a "fast filtering" algorithm, which incorporates the statistics of the observation process as initial conditions for a dynamical system, this question is analyzed in terms of the phase portrait of a "universal" nonlinear dynamical system. This point of view has additional advantages as well, since it enables one to use the theory of dynamical systems to study the sensitivity of the Kalman filter to (small) changes in initial conditions; e.g., to changes in the statistics of the underlying process. This is especially important since these statistics are often either approximated or estimated. In this paper, for a scalar observation process, necessary and sufficient conditions for the Kalman filter to converge are derived using methods from stochastic systems and from nonlinear dynamics---especially the use of stable, unstable, and center manifolds. It is also shown that, in nonconvergent cases, there exist periodic points of every period $p$, $p\ge 3$ that are arbitrarily close to initial conditions having unbounded orbits, rigorously demonstrating that the Kalman filter can also be "sensitive to initial conditions."

Modeling of PRML timing loop as a Kalman filter

Abstract: This paper demonstrates the equivalence between the timing loop used in the partial response maximum likelihood (PRML) disk-drive channel and a modified Kalman filter. The Kalman filter is shown to reach a steady-state gain that corresponds to the timing loop when in its steady-state tracking mode. The dependence of the steady-state gain on the noise and physical properties of the loop is investigated. The timing loop is then optimized using the properties of the Kalman filter.

The design of a fully active suspension system incorporating a Kalman filter for state estimation

Abstract: Concerns the use of a Kalman filter for a fully active road vehicle suspension system. The usual detectability and stabilizability conditions must be satisfied and all the noise signals must be white. The detectability condition is obeyed by using the filtered white noise road input. The use of this input, as opposed to the frequency limited integrated white noise input, is justified by r.m.s. simulation results. The stabilizability condition is obeyed by having a spring in parallel with the actuator, which is also of practical significance for suspending the static load, hence decreasing actuator power consumption. The description of the noise processes as bandlimited white noise models a worst case scenario as all of the noise signal is present in the operational bandwidth of the closed loop system. The closed loop system using the Kalman filter was simulated and compared to that using full state feedback. Results using the Kalman filter were encouraging, showing a small degradation in performance compared to the nominal system. Interesting results were obtained for road roughnesses different from those for which it was designed. There was a very small degradation in performance, which indicates that there seems to be no need to adapt the Kalman filter gain for different road conditions. Therefore, the potential improvements of using this system, as opposed to the usual LQG method using full state feedback, are enormous. However, it was found that the use of the Kalman filter led to a marked degradation in the stability margins of the system.< >

Modified extended Kalman filtering

Abstract: In this paper we compare the performance of the extended Kalman filter (EKF) with that of a proposed filter called modified extended Kalman filter (MEKF), in terms of the actual state and the estimated state as a function of time and the integral-squared error as a function of noise intensity both for dynamics and measurement noise. All the simulation results indicate that the performance of the MEKF is better than EKF. For illustration, numerical examples are presented using both EKF and MEKF. >

General decentralized Kalman filters

Abstract: This paper considers a general decentralised Kalman filter, unique because it: requires no fusion center yet allows the different estimators to employ distributed models; minimizes communication with respect to message size and topology; requires no explicit knowledge of the transformations between the estimators; and reduces to previous results when appropriately constrained.

A method for improving extended Kalman filter performance for angle-only passive ranging

Abstract: A relatively simple method is presented which eliminates previously reported (Oct. 1985) erratic estimation performance associated with Cartesian formulations of the extended Kalman filter (EKF) for the 2D angle-only emitter location problem. The technique is based on an initialization procedure which combines a priori probability density function (pdf) information with single measurement a posteriori pdf information in a manner which is more efficient than the EKF. Simulation results are presented which demonstrate the utility of the technique as compared with a previously offered modified gain EKF. >

New exact nonlinear filters: theory and applications

Abstract: A new exact recursive filter is derived for nonlinear estimation problems. The new nonlinear theory includes the Kalman filter as a special case. This filter is practical to implement in real- time applications, and it has a computational complexity that is comparable to the Kalman filter. The measurements are made in discrete time, but the random process to be estimated evolves in continuous time.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A square root filter algorithm for dynamic state estimation of electric power systems

Abstract: The proper analysis and successful operation of a power system implies a reliable estimate of its state. As a result state estimation (SE) is nowadays considered as the heart of modern control centers. Several dynamic state estimation algorithms based on the extended Kalman filtering theory (EKF) have been proposed in the literature but, as has been reported, numerical problems may arise in its implementation in practice. To circumvent the problems inherent to the Kalman filter algorithm the authors present in this paper an algorithm for power system state estimation based on the square root filtering technique. The estimator proposed is decoupled in nature due to modifications made in the measurement equations resulting in a new observation model. >

A SVD-based extended Kalman filter and applications to aircraft flight state and parameter estimation

Abstract: In this paper, a new robust extended Kalman filtering algorithm based on singular value decomposition (SVD) of a covariance/information matrix is presented with application to the flight state and parameter estimation of aircraft. The presented algorithm not only has a good numerical stability but also can handle correlated measurement noise without any additional transformation. The algorithm is formulated in the form of vector-matrix operations, so it is also useful for parallel computers. The applications to the flight state and parameter estimation by simulated and actual flight test data computation of two types of Chinese aircraft show that the new algorithm presented in this paper can give more accurate estimates of flight state and parameters than an extended Kalman filter (EKF) for different initial values and noise statistics. Moreover, the new algorithm has less requirements for maneuvering shapes, noise levels, data length and better convergency than those of the EKF. The computational requirements for one-step filtering updates of the new filter have been reduced greatly by exploiting some special features of system and measurement models. It is proved that the new filtering algorithm can give good results even for low sample rate flight test data.

Node decoupled extended Kalman filter based learning algorithm for neural networks

Abstract: The use of extended Kalman filter (EKF) is common in estimation of nonlinear system parameters. It has also found application in training of feedforward neural networks. A heuristic modification of the EKF algorithm known as the node decoupled EKF (NDEKF) algorithm, which improves upon the EKF algorithm by significantly reducing computation time and memory requirements, appears very promising. The purpose of this paper is to present the NDEKF algorithm in a form suitable for coding readily into a computer program. Matlab implementation of the algorithm with simulation examples is included. >

Nonlinear information space: a practical basis for decentralization

Abstract: In this paper the Nonlinear Information Filter is derived from the Extended Kalman Filter. A nonlinear system is considered. Linearizing the state and observation equations, a linear estimator which keeps track of total state estimates is conceived; the Extended Kalman Filter. The linearized parameters and filter equations are expressed in information space. This gives a filter that predicts and estimates information about nonlinear state parameters given nonlinear observations and nonlinear system dynamics. The Nonlinear Information Filter derivation is contrasted to that of the Linear Information filter. Pitfalls of a naive extension of the later to the former are thus identified. Furthermore, the Nonlinear Information filter is decentralized and distributed, to give the Distributed and Decentralized Nonlinear Information. Application is real decentralized data fusion and distributed control is proposed. Specifically, realtime distributed/decentralized control of a navigating, modular wheeled robot is considered.

Application of the Kalman filter to computational problems in statistics.

Abstract: Publisher Summary This chapter discusses the application of the Kalman filter to computational problems in statistics The Kalman filter is a linear filtering algorithm developed to solve estimation and control problems in engineering such as monitoring the position and velocity of a satellite orbiting the earth from signals received at ground tracking stations The chapter illustrates its use to evaluate a Gaussian likelihood where the observational error process is Gaussian serially correlated noise The problem of modeling human biological rhythm data is discussed in this chapter Core temperature data is an often studied biological rhythm used to estimate the properties of the human biological clock The sequential manner in which the Kalman filter can be used to evaluate Gaussian likelihoods has led to an efficient technique for handling missing data in a time series estimation problem The Kalman filter can also be used to compute posterior densities for Bayesian statistical models The chapter demonstrates the use of the Kalman filter for evaluating a Gaussian likelihood as part of the expectation and maximization (EM) algorithm

Demodulation of noisy phase or frequency modulated signals with Kalman filters

Abstract: The paper addresses the problem of estimating the argument of a phase or frequency modulated sinusoidal signal of unknown amplitude in the presence of additive Gaussian white noise of known variance. It is assumed that a state space model for both the unknown phase and the unknown amplitude is available either as an exact or approximate description of the reality. The problem is solved by an extended linearised Kalman filter. The basic idea of using extended Kalman filters was previously described but has not been used to perform simultaneous amplitude and phase demodulation. >