Showing papers on "Invariant extended Kalman filter published in 1986"

Numerical aspects of different Kalman filter implementations

Abstract: A theoretical analysis is made of the error propagation due to numerical roundoff for four different Kalman filter implementations: the conventional Kalman filter, the square root covariance filter, the square root information filter, and the Chandrasekhar square root filter. An experimental analysis is performed to validate the new insights gained by the theoretical analysis.

Exact finite-dimensional nonlinear filters

Abstract: A new nonlinear filter is derived for continuous-time processes with discrete-time measurements. The filter is exact, and it can be implemented in real time with a computational complexity that is comparable to the Kalman filter. This new filter includes both the Kalman filter and the discrete-time version of the Benes filter as special cases. Moreover, the new theory can handle a large class of nonlinear estimation problems that cannot be solved using the Kalman or discrete-time Benes filters. A simple approximation technique is suggested for practical applications in which the dynamics do not satisfy the required conditions exactly. This approximation is analogous to the so-called "extended Kalman filter" [10], and it represents a generalization of the standard linearization method.

Toward the Application of the Kalman Filter to Regional Open Ocean Modeling

Abstract: A partial differential equation model is defined for ocean meteorological prediction and synoptic analysis. The Kalman filter used for data assimilation is described and applied to the one-dimensional linear barotropic quasi-geostrophic model with periodic and open boundary conditions. The model accounts for eddy scale dynamics in the ocean. The assumptions made in the forecast model are discussed, along with comparisons of the error variances expected with the filter and from an objective analysis method. The effectiveness of the Kalman filter is demonstrated and subsequent efforts to extend the filter to two dimensions are indicated.

Efficient Approximation of Kalman Filter for Target Tracking

Abstract: A Kalman filter in the Cartesian coordinates is described for a maneuvering target when the radar sensor measures range, bearing, and elevation angles in the polar coordinates at high data rates. An approximate gain computation algorithm is developed to determine the filter gains for on-line microprocessor implementation. In this approach, gains are computed for three uncoupled filters and multiplied by a Jacobian transformation determined from the measured target position and orientation. The algorithm is compared with the extended Kalman filter for a typical target trajectory in a naval gun fire control system. The filter gains and the tracking errors for the proposed algorithm are nearly identical to the extended Kalman filter, while the computation requirements are reduced by a factor of four.

Application of an extended Kalman filter for state estimation of a yeast fermentation

Abstract: The development of a Kalman filter for state and parameter estimation of a biotechnical process is discussed Because of the large complexity of biotechnical processes, mathematical models for online estimation are based on extensive simplifications Therefore model errors in the structure and parameters cannot be avoided In such situations, simulations of the process in combination with the estimator are very helpful during the design phase: these permit fast examinations of the different behaviour of linear filters compared to nonlinear algorithms and also investigations of the influence of sampling interval and initial values of state and filter variables on the estimation By the use of such simulations, the suitability of process models with various degrees of simplifications can also be easily tested Based on the simulations, an extended Kalman filter with iteration of the output equations was chosen Besides the states, two parameters of a third order process model are estimated online The filter algorithm was tested during batch processes and worked well after a slight modification The filter behaviour observed in the experiments was very similar to the simulations

Storm surge prediction using Kalman filtering

Abstract: In this study the theory of Kalman filtering has been employed to develop a new method for predicting water-levels along the Dutch coast. The combination of the standard Kalman filter with a non-linear tidal model of the entire North Sea is, from a computational point of view, not (yet) feasible. Therefore, in this investigation two different approaches have been developed. The first is based on the approximation of the tidal movement in the Dutch coastal area by a one-dimensional model. The two-dimensional effects due to the wind and the Coriolis force are taken into account by introducing some additional, empirical equations. The finite difference scheme and the system noise processes, introduced to describe the uncertainty associated with the model, are chosen such that numerical difficulties are avoided. Water-levels and velocities as well as the uncertain parameters in the model are estimated on-line by the Kalman filter. Since the model is continuously being adapted to the changing conditions, even this simple conceptual model gives satisfactory predictions. However, the time interval over which accurate predictions can be produced is limited because the one-dimensional approximation is only realistic for a smal1 part of the southern North Sea. To increase the prediction interval the second Kalman filter approach that is developed in this investigation is based on a two-dimensional model of the entire North Sea. The extension of the one-dimensional filter to two space dimensions does not give rise to conceptual problems but, as noted before, impose an unacceptably greater computational burden. In order to reduce this burden, the Kalman filter is approximated by a time-invariant one. In this case the time-consuming filter equations do not have to be computed over again as new measurements become available, but need only be solved once. Furthermore, by defining the system noise processes on a coarse grid and by employing a Chandrasekhar-type filter algorithm; a computationally attractive implementation of the filter is obtained. It is shown that the algorithm can be vectorized efficiently and that using a CDC CYBER 205 vector processor it is possible to combine the steady-state filter approach with very large models. Numerical difficulties can be avoided by carefully choosing the finite difference scheme, the boundary treatment and most important, the system noise processes. The filter has been tested extensively using simulated data as well as field data. The results show excellent filter performance, especially if we take into account that the number of measurements available (as yet) has been very limited. With respect to the results of the deterministic model without using tbe water-levels measurements available, the improvement obtained by filtering these measurements is substantial.

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.

VLSI Implementation of real-time Kalman filter

Abstract: In this paper, the problem of parallel implementation of the square-root Kalman filters is addressed. In the system level, our approach is to apply systolic type processor arrays as basic building blocks to speed up the matrix operations required in each iteration. Specifically, by utilizing a sparse matrix structure, we derive a simple systolic array configuration which is able to solve a rotation operation very efficiently. To maximize the parallelism, we also exploit an inter-array pipelining scheme through the overlapping of execution between successive processor arrays. As a result, several modules can be tightly coupled to form a dedicate Kalman Filter processor for real time applications. We estimate that with O(n2) processors, it would take O(4n+3r-3) time units to complete one Kalman filter iteration, where n is number of states and r is number of inputs.

New Nonlinear Filters and Exact Solutions of the Fokker-Planck Equation

Abstract: A new nonlinear filter is derived for continuous time processes with discrete time measurements. The filter is exact, and it can be implemented in real-time with a computational complexity that is comparable to the Kalman filter. This new filter includes both the Kalman filter and the discrete time version of the Bene? filter as special cases. Moreover, the new theory can handle a large class of nonlinear estimation problems that cannot be solved using the Kalman or discrete time Bene? filters. A new approximation technique is suggested for problems that do not satisfy the theoretical conditions exactly. This approximation is simple and straightforward, analogous to the extended Kalman filter.

On the sensitivity of a discrete-time Kalman filter to plant-dynamics modelling errors

Abstract: The sensitivity of the discrete-time Kalman filter to errors in the state transition matrix is examined. The approach makes use of the matrix Taylor expansion and estimates the deviation from the ‘nominal’ computed performance. Although only the first term of the expansion is retained, satisfactory accuracy is achieved. The method is considerably simple and constitutes a quick way to diagnose or establish limits for divergence. Direct application to stochastic observers is also possible

State Estimation of a Scalar Dynamic Precipitation Model From Time‐Aggregate Observations

Abstract: Developed is an exact, recursive, state estimator for linear, scalar systems whose observables are time integrals, over an interval Δt, of linear functions of the state. In general, implementation of the estimator results in excessive computations. The computationally inexpensive, ordinary Kalman filter with inflated observation error variance is used to approximate the developed estimator in terms of prediction accuracy. Comparison of the exact estimator and of the Kalman filter, using precipitation data of various aggregation intervals Δt and a scalar dynamic precipitation model, shows that the Kalman filter develops prediction bias as Δt increases.

A Factorized Extended Kalman Filter

Abstract: Kalman filtering represents formidable computational linear algebra requirements for each new input measurement vector. The air-to-air missile guidance problem is addressed for which an extended Kalman filter (EKF) is required because the measurements are nonlinear in Cartesian coordinates. An explicit formulation is used. At the outset, we discretize the system dynamics and measurement model and incorporate a discrete-time EKF. A factorized L D LT algorithm is used to propagate the covariance matrices between sample times. A simulation analysis of the number of data bits required in the computations is provided. Comparison with other EKF algorithms shows that this method requires only 18 bit accuracy (compared to 32-40 bits for other methods). Quantitative position, velocity and acceleration estimates obtained for a highly maneuverable target are presented. A high-accuracy floating point optical processor is presented that is capable of computing the full EKF to allow a new measurement update each msec.

The Kalman filter: an introduction to the mathematics of linear least mean square recursive estimation

Abstract: This paper introduces the fundamental ideas of Kalman filtering, a recursive estimation technique widely used for continuous estimation of the state of a dynamic system. The estimation problem is posed within the well known Hilbert space framework of classical linear analysis. This permits an easily grasped geometric interpretation which is stripped of cumbersome details that tend to obscure the essential notions. Considerable emphasis is placed on the development of the mathematical models for the state and measurement equations. A practical example of a real‐world dynamic system (the motion of a ship) is used to motivate the form of the state equation required in Kalman filtering, as well as the measurement equation. The recursive estimator and error covariance equations are derived for a one‐dimensional dynamic system using a sequence of geometric visualizations as the derivation proceeds. The more tedious algebraic manipulations, which are not needed for an essential understanding of the derivation, a...

Accuracy of the decoupled Kalman filter

Abstract: An explicit analytical solution is obtained for the suboptimal covariance matrix of a decoupled Kalman filter. The result is used to determine when filter decoupling breaks down.

Kalman filter parameter identification: a practical approach:

Abstract: This paper describes the principles of application of the extended Kalman filter identification technique as a means of identification of linear systems with Gaussian random inputs. Major consideration is given to an algorithmic implementation rather than to theoretical background in an attempt to make the technique more widely available to the engineer. Results of simulation studies suggest guidelines to aid in successful application of the technique to experimental situations.

Estimation of atmospheric CO2 concentration using Kalman filtering

Abstract: Using a dynamic state model for the observed upward trend and sinusoidal variation, a Kalman filter is constructed to estimate the atmospheric CO2 concentration, The process noise is assumed to be white with an unknown covariance, so an adaptive scheme is used to estimate the steady-state Kalman gain matrix. Several tests for optimality are performed on the adaptive filter. Measured data are then filtered using the Kalman algorithm. The filtering results are shown to reduce the variability of the airborne fraction of fossil-fuel-produced atmospheric CO2.

Factorized extended Kalman filter for optical processing.

Abstract: Kalman filtering represents formidable linear algebra computational requirements for each new input measurement vector. An architecture-motivated implementation of a discrete-time extended Kalman filter algorithm is presented. This particular formulation takes advantage of the following features of the optical processor architecture: the ability to perform matrix–vector operations, floating-point capabilities, and specially designed matrix–vector L U decomposition operations. A factorized L D LT algorithm is used to propagate the covariance matrices between sample times. The air-to-air missile guidance problem is used as a case study wherein an extended Kalman filter is required due to the nonlinear nature of the measurement equations.

A fixed-complexity nonlinear estimation technique

Abstract: The typical nonlinear estimation problem is solved by a quasi-linear technique, the extended Kalman filter (EKF). The EKF provides satisfactory results if first order approximations to the system equations are adequate. If they are not and the filtering time is long, substantial estimation errors may build up. To overcome this problem, various techniques have been developed, e.g., "second-order filters" and "polynomial filters" have been proposed1-3. In this paper, the nonlinear system and observation equations are linearized, the higher order terms in the Taylor series are treated as unknown time functions and these functions are estimated using a block sequential least squares technique. An extended observer is then used to solve for the linear perturbation from the nominal state and a second estimator is used to estimate the residual observer error.

A recursive solution for a fading memory filter derived from Kalman filter theory

Abstract: A simple recursive solution for a class of fading memory tracking filters is presented. A fading memory filter provides estimates of filter states based on past measurements, similar to a traditional Kalman filter. Unlike a Kalman filter, an exponentially decaying weight is applied to older measurements, discounting their effect on present state estimates. It is shown that Kalman filters and fading memory filters are closely related solutions to a general least squares estimator problem. Closed form filter transfer functions are derived for a time invariant, steady state, fading memory filter. These can be applied in loop filter implementation of the Deep Space Network (DSN) Advanced Receiver carrier phase locked loop (PLL).

Design of Linear Quadratic Regulators and Kalman Filters

Abstract: AESOP solves problems associated with design of controls and state estimators for linear time-invariant systems. Systems considered are modeled in state-variable form by set of linear differential and algebraic equations with constant coefficients. Two key problems solved by AESOP are linear quadratic regulator (LQR) design problem and steady-state Kalman filter design problem. AESOP is interactive. User solves design problems and analyzes solutions in single interactive session. Both numerical and graphical information available to user during the session.