Showing papers on "Invariant extended Kalman filter published in 1996"

Introduction to random signals and applied Kalman filtering : with MATLAB exercises and solutions

Abstract: Probability and Random Variables: A Review. Mathematical Description of Random Signals. Response of Linear Systems to Random Inputs. Wiener Filtering. The Discrete Kalman Filter, State-Space Modeling, and Simulation. Prediction, Applications, and More Basics on Discrete Kalman Filtering. The Continuous Kalman Filter. Smoothing. Linearization and Additional Intermediate-Level Topics on Applied Kalman Filtering. More on Modeling: Integration of Noninertial Measurements Into INS. The Global Positioning System: A Case Study. Appendices. Index.

A moving horizon-based approach for least-squares estimation

Abstract: A general formulation of the moving horizon estimator is presented. An algorithm with a fixed-size estimation window and constraints on states, disturbances, and measurement noise is developed, and a probabilistic interpretation is given. The moving horizon formulation requires only one more tuning parameter (horizon size) than many well-known approximate nonlinear filters such as extended Kalman filter (EFK), iterated EKF, Gaussian second-order filter, and statistically linearized filter. The choice of horizon size allows the user to achieve a compromise between the better performance of the batch least-squares solution and the reduced computational requirements of the approximate nonlinear filters. Specific issues relevant to linear and nonlinear systems are discussed with comparisons made to the Kalman filter, EKF, and other recursive and optimization-based estimation schemes.

Mapping tropical Pacific sea level : Data assimilation via a reduced state space Kalman filter

Abstract: The well-known fact that tropical sea level can be usefully simulated by linear wind driven models recommends it as a realistic test problem for data assimilation schemes. Here we report on an assimilation of monthly data for the period 1975-1992 from 34 tropical Pacific tide gauges into such a model using a Kalman filter. We present an approach to the Kalman filter that uses a reduced state space representation for the required error covariance matrices. This reduction makes the calculation highly feasible. We argue that a more complete representation will be of no value in typical oceanographic practice, that in principle it is unlikely to be helpful, and that it may even be harmful if the data coverage is sparse, the usual case in oceanography. This is in part a consequence of ignorance of the correct error statistics for the data and model, but only in part. The reduced state space is obtained from a truncated set of multivariate empirical orthogonal functions (EOFs) derived from a long model run without assimilation. The reduced state space filter is compared with a full grid point Kalman filter using the same dynamical model for the period 1979-1985, assimilating eight tide gauge stations and using an additional seven for verification (Miller et al., 1995). Results are not inferior to the full grid point filter, even when the reduced filter retains only nine EOFs. Five sets of reduced space filter assimilations are run with all tide gauge data for the period 1975-1992. In each set a different number of EOFs is retained: 5, 9, 17, 32, and 93, accounting for 60, 70, 80, 90, and 99% of the model variance, respectively. Each set consists of 34 runs, in each of which one station is withheld for verification. Comparing each set to the nonassimilation run, the average rms error at the withheld stations decreases by more than 1 cm. The improvement is generally larger for the stations at lowest latitudes. Increasing the number of EOFs increases agreement with data at locations where data are assimilated; the added structures allow better fits locally. In contrast, results at withheld stations are almost insensitive to the number of EOFs retained. We also compare the Kalman filter theoretical error estimates with the actual errors of the assimilations. Features agree on average, but not in detail, a reminder of the fact that the quality of theoretical estimates is limited by the quality of error models they assume. We briefly discuss the implications of our work for future studies, including the application of the method to full ocean general circulation models and coupled models.

H∞ nonlinear filtering

Abstract: This paper investigates the problem of H∞ estimation of nonlinear processes. An estimator, which may be nonlinear, is looked for so that a given bound on the ratio between the energy of the estimation error and the energy of the oxogeneous inputs to the estimated process is achieved. Conditions for the existence of such an estimator and formulas for its derivation are obtained using both the game theory approach and the theory of dissipative systems. The results of the paper extend the recent results on H∞ nonlinear control. They are demonstrated by a simple example of a linear system with a nonlinear measurement rule and compared with corresponding results that are obtained by the extended Kalman filter.

The Kalman filter in finance

Abstract: Preface. 1. Introduction. 2. Test for parameter stability. 3. Flexible Least Squares. 4. The Kalman filter. 5. Parameter estimation. 6. The estimates, reconsidered. 7. Modeling with the Kalman filter. A. Tables of references. B. The programs and the data. Bibliography. Index.

Incremental Least Squares Methods and the Extended Kalman Filter

Abstract: In this paper we propose and analyze 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. We focus on the extended Kalman filter, which may be viewed as an incremental version of the Gauss--Newton method. We provide a nonstochastic analysis of its convergence properties, and we discuss variants aimed at accelerating its convergence.

A new motor speed estimator using Kalman filter in low-speed range

Abstract: In this paper, a new machine drive technique using novel estimation strategy for the very low-speed operation to estimate both the instantaneous speed and disturbance load torque is proposed. In the proposed algorithm, a Kalman filter is incorporated to estimate both the motor speed and the disturbance torque. The Kalman filter is an optimal state estimator and is usually applied to a dynamic system that involves a random noise environment. The effects of parameter variations are discussed, and it is verified that the system is stable to the modeling error. Experimental results confirm the validity of the proposed estimation technique.

A new technique for nonlinear estimation

Abstract: A new nonlinear filter referred to as the state-dependent Riccati equation filter (SDREF) is presented. The SDREF is derived by constructing the dual of a little known nonlinear regulator control design technique which involves the solution of a state-dependent Riccati equation (SDRE) and which has been appropriately called the SDRE control method. The resulting SDREF has the same structure as the continuous steady-state linear Kalman filter. In contrast to the linearized Kalman filter (LKF) and the extended Kalman filter (EKF) which are based on linearization, the SDREF is based on a parameterization that brings the nonlinear system to a linear structure having state-dependent coefficients (SDC). In a deterministic setting, before stochastic uncertainties are introduced, the SDC parameterization fully captures the nonlinearities of the system, It was shown in Cloutier et al. (1996) that, in the multivariable case, the SDC parameterization is not unique and that the SDC parameterization itself can be parameterized. This latter parameterization creates extra degrees of freedom that are not available in traditional filtering methods. These additional degrees of freedom can be used to either enhance filter performance, avoid singularities, or avoid loss of observability. The main intent of this paper is to introduce the new nonlinear filter and to illustrate the behaviorial differences and similarities between the new filter, the LKF, and the EKF using a simple pendulum problem.

Dual Kalman Filtering Methods for Nonlinear Prediction, Smoothing and Estimation

Abstract: Prediction, estimation, and smoothing are fundamental to signal processing. To perform these interrelated tasks given noisy data, we form a time series model of the process that generates the data. Taking noise in the system explicitly into account, maximum-likelihood and Kalman frameworks are discussed which involve the dual process of estimating both the model parameters and the underlying state of the system. We review several established methods in the linear case, and propose several extensions utilizing dual Kalman filters (DKF) and forward-backward (FB) filters that are applicable to neural networks. Methods are compared on several simulations of noisy time series. We also include an example of nonlinear noise reduction in speech.

The application of Kalman smoother theory to the estimation of 4DVAR error statistics

Abstract: Modern atmospheric data assimilation theory is dominated by the four-dimensional variational (4DVAR) and Kalman filter/smoother approaches. Both generate analysis weights (explicitly or implicitly) which are dynamically determined by the assimilation model. A Kalman smoother is basically a generalization of the Kalman filter which can process future observations. In control theory, a generalization of 4DVAR called Pontryagin optimization can account for an imperfect assimilating model. Pontryagin optimization and the fixed-interval Kalman smoother are equivalent when both methods use the same statistical information. We use the equivalence between Pontryagin optimization and the Kalman smoother to examine the effect of the perfect model assumption on the error statistics and analysis weights of the 4DVAR algorithm. This is done by developing the Kalman smoother equations for a very simple assimilating model. A procedure for diagnosing the effect of model error, based on the observational cost function, is also developed. DOI: 10.1034/j.1600-0870.1996.t01-1-00003.x

Best choice of coordinate system for tracking coordinated turns

Abstract: A standard approach to tracking is to use the extended Kalman filter (EKF) applied to a nonlinear state-space model. We compare two conceivable choices of state variables for modeling civil aircrafts. One where Cartesian velocities are used and one where absolute velocity and heading angle are used. In both choices, Cartesian coordinates are used for position and angular velocity for turning. It is shown that the latter state vector always performs better. This is proven by considering the linearization error made in the extended Kalman filter applied either to a time-continuous model or a discretized model. The result is supported by a Monte Carlo simulation study.

An extended Kalman filter frequency tracker for high-noise environments

Abstract: The problem of constructing a frequency tracker for weak, narrowband signals with slowly varying frequency is considered. An extended Kalman filter is proposed that uses prior knowledge of the nature of the signal to overcome the difficulties presented by the inherent nonlinearity of the problem and the very low signal-to-noise ratios.

Optimal Recursive Iterative Algorithm for Discrete Nonlinear Least-Squares Estimation

Abstract: The estimation algorithm developed offers an alternative to standard recursive nonlinear estimators such as the extended Kalman filter and the iterated extended Kalman filter. The algorithm, which is developed from a quadratic cost function basis, splits the problem of cost function minimization into a linear first step and a nonlinear second step by defining new first-step states that are nonlinear combinations of the unknown states. Estimates of the firststep states are obtained by minimizing the first-step cost function using a Kalman filter formulation. Estimates of the unknown, or second-step, states are obtained by minimizing the second-step cost function using an iterative Gauss-Newton algorithm. The two-step estimator is shown to be optimal for static problems in which the time variation of the measurement equation can be separated from the unknowns. This method is then generalized by approximating the nonlinearity as a perturbation of the dynamic update, while keeping the measurement cost function the same. In contrast, the extended Kalman filter and the iterated extended Kalman filter linearize the measurement cost function, resulting in suboptimal estimates. Two example applications confirm these analytical results.

Elastically Adaptive Deformable Models

Abstract: We present a novel technique for the automatic adaptation of a deformable model's elastic parameters within a Kalman filter frame-work for shape estimation applications. The novelty of the technique is that the model's elastic parameters are not constant, but time varying. The model for the elastic parameter variation depends on the local error of fit and the rate of change of the error of fit. By augmenting the state equations of an extended Kalman filter to incorporate these additional variables and take into account the noise in the data, we are able to significantly improve the quality of the shape estimation. Therefore, the model's elastic parameters are initialized always to the same value and they subsequently modified depending on the data and the noise distribution. In addition, we demonstrate how this technique can be parallelized in order to increase its efficiency. We present several experiments to demonstrate the effectiveness of our method.

A map matching method with the innovation of the kalman filtering

Abstract: This paper proposes a map-matching method for automotive navigation systems The proposed method utilizes a Kalman filter algorithm and can achieve more accurate positioning than the correlation method The performance of the algorithm is verified by simulation

Dynamic rectification of data via recurrent neural nets and the extended Kalman filter

Abstract: The presence of autocorrelated measurement errors and/or measurement bias in process measurements poses serious problems in the rectification of data taken from dynamic processes. The proposed procedure to resolve these problems involves the use of recurrent neural networks (RNN) and the extended Kalman filter (EKF). By interpreting RNNs within a nonlinear state-space context, a state-augmented EKF can be used to optimally estimate both the states of the RNNs and noise and bias models. RNN models can be identified off-line and utilized for data rectification within the extended Kalman filter in process environments in which badly autocorrelated measurement errors exist in the data. The same technique is also used to estimate measurement bias present in both process input and output variables. This approach has the advantage that models developed from “first principles” are not required and that rectification can be performed solely on the basis of the contaminated dynamic data.

Dead reckoning navigation of a mobile robot using an indirect Kalman filter

Abstract: An encoder and a gyroscope fusion algorithm for an autonomous mobile robot (AMR) navigation using indirect Kalman filter scheme is proposed. An encoder based navigation algorithm is developed and based on the navigation algorithm, a navigation error model is derived by linear perturbation. The indirect Kalman filter for fusion algorithm is realized by applying Kalman filter to the navigation error model. Furthermore the use of indirect feedback Kalman filter, which feeds back the error estimates to the main navigation algorithm is proposed for the AMR navigation. The simulation is performed and the results show that the navigation algorithm with the indirect feedback Kalman filter can successfully combine the encoder and gyroscope measurements and estimate the position and heading angle of the AMR.

Kalman filter ionospheric delay estimator

Abstract: An estimator structure include a nonlinear modified Kalman filter for initial estimate and plurality of bias estimators and a parallel bank of Kalman filters for ambiguity resolution and providing final estimates. The estimator structure is useful for real-time computation of ionospheric propagation delay introduced in the navigation signals by dual frequency GPS receivers.

Tutorial: The Likelihood Interpretation of the Kalman Filter.

Abstract: 1 This is only an extract from this work and contains references to material from other chapters in the larger document.

Order statistic fast Kalman filter

Abstract: This paper deals with the derivation of a new adaptive filtering algorithm, which takes into consideration the often encountered case of impulsive perturbations. The proposed method is a combination of a RLS-based algorithm and the Order Statistic (OS) filter, and can be seen as an extension of the LMS-type Order Statistic filter (OSLMS). In order to obtain a computational burden comparable to the OSLMS filter, our derivation is based on a fast version of the RLS algorithm-the Fast Kalman filter. We will show that the new algorithm is not a coarse extension of the OSLMS filter and that care should be taken when performing the order statistic filtering operation.

Optimal robust filtering with time-varying parameter uncertainty

Abstract: Stochastic linear systems subject to time-varying parameter uncertainties affecting both system dynamics and noise statistics are considered. A linear filter is used to estimate a linear combination of the states of the system. When the filter is given, the stabilizing solution of a suitable Riccati equation is shown to yield an upper bound for the covariance of the estimator error. The main problem addressed in the paper is the design of an ‘optimal robust filter’ that minimizes such a covariance bound. Necessary conditions are given for the existence of an optimal reduced-order robust filter as well as necessary and sufficient conditions for the full-order case. The computation of the optimal filter calls for the solution of a Riccati equation that generalizes the standard Riccati equation for the Kalman filtering problem. A numerical example is provided in which the new filter is compared with both the Kalman and H ∞-filters.

Initialization of the kalman filter with partially diffuse initial conditions

Abstract: The problem of computing estimates of the state vector when the Kalman filter is seeded with an arbitrarily large variance is considered To date the response in the literature has been the development of a number of relatively complex hybrid filters, usually involving additional quantities and equations over and above the conventional filter We show, however, that a certain square root covariance filter is capable of handling the complete range of variances (zero, positive and infinite) without modification to the filtering equations themselves and without additional computation loads Instead of the more conventional Cholesky factorization, our filter employs an alternative matrix factorization procedure based on a unit lower triangular matrix and a diagonal matrix This permits the use of a modified form of fast Givens transformations, central to the development of an efficient algorithm

A Kalman filter to estimate direction for automotive navigation

Abstract: A Kalman filter algorithm to estimate direction for automotive navigation is reported. An extended Kalman filter is used to combine a magnetic compass and a rate gyroscope and to compensate for the sensor errors; the result is an optimal estimate for heading direction. A mathematical model for magnetic compass errors caused by body magnetization and the body effect of magnetic material is proposed. An error model of the rate gyroscope is also established. Errors of both sensors are calibrated through a computer simulation. Finally, experimental navigation results are demonstrated.

Nonlinear filters based on taylor series expansions

Abstract: We have been dealing with the linear transition and measurement equations. The equations derived from economic theories, however, are nonlinear in general. Unless the distributions are normal and the measurement and transition equations are linear, we cannot derive the explicit expression for the filtering algorithm. Therefore, some approximation is necessary for estimation. In Chapter 3, the nonlinear filtering algorithms are derived by approximating the nonlinear measurement and transition equations, based on the Taylor series expansion.

Modification of the extended Kalman filter with an additive term of instability

Abstract: In this paper we propose an observer for nonlinear systems similar to the extended Kalman filter. The observer gain is computed by a Riccati differential equation assuming a more instable system. The purpose of this modification is two-fold: first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities.

A novel Kalman filter-based navigation using beacons

Abstract: A novel Kalman filter-based scheme is proposed to estimate the navigational states of a vehicle from the range measurements obtained using beacons (transmitter-receivers): at least three placed at known locations and one on the vehicle. The novelty of the scheme stems from 1) the use of pseudomeasurements which are some nonlinear function of the range measurements so that the measurement model is linear with the resulting Kalman filter globally convergent, 2) the measurements of the pair-wise sums and differences in the ranges so as to reduce the sensitivity to the measurement noise, and 3) the optimal location of the beacons to ensure that the state estimation error is minimised, resulting in the estimation accuracy being linear in the true states. Both the bias and the random errors are considered to account for the sources of errors such as the sensor bias, the faults, and the sensor noise. The proposed scheme is evaluated on a number of simulated examples.

Kalman filter with hypothesis testing: A tool for estimating uncertain parameters

Abstract: In this paper we present a simple and practical algorithm for the estimation of uncertain parameters of linear systems. The uncertainty is twofold, involving random observation noise, and possible jumps in the parameter values. The jumps may occur at unknown points in time, and are of unknown magnitudes and directions. The algorithm is based on the Kalman filter, with a single-sample hypothesis test, which is used to employ a three-state decision rule (yes, no, maybe). The "maybe" choice invokes a fading memory Kalman filter. The overall algorithm contains the constant parameter filter, fading memory filter, and the set of tests and rules that enable it to switch back and forth between the two filters. Application examples are presented.