Showing papers on "Alpha beta 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.

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 variable update rate algorithm for tracking targets with a phased array radar

Abstract: The analysis postulates an algorithm for adaptively varying the target-track update rate as a function of target manoeuvring. The algorithm proposes to vary the target observation time interval by an amount that tends to maintain a constant residual error. As an example, the algorithm is shown to be implemented with a two-dimensional alpha-beta filter. Simulated results are calculated for a target manoeuvre consisting of a single 90° turn. The results show that when a manoeuvre is detected, the sampling time interval can be reduced rapidly to facilitate tracking with reduceed error. After the manoeuvre, the sampling interval is increased automatically so more time can be made available to the phased array radar for other activities.

Minimal-order observer designs for linear time-varying multivariable systems

Abstract: This note presents two simple approaches for the design of minimal-order observer for linear time-varying systems. Both techniques rely on canonical transformations of the given system. As a consequence, simple expressions are obtained which permit the observer constraint equation to be solved effectively. The observers possess the desirable property of having a time-invariant observer coefficient matrix with arbitrary eigenvalues. A comparison is made between the proposed methods from which conclusions are drawn. An illustrative example is also included.

Modal approach to parametric state observer design

Abstract: It is shown that in the design of an observer for asymptotic reconstruction of the state of a linear time-invariant system the gain matrix can be explicitly parameterised by two sets of design parameters which in common completely characterise the modal behaviour of a state observer. This modal approach is well suited both for further investigations of state observer properties and for the development of practical observer design techniques.

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.

State estimation using continuous orthogonal functions

Abstract: A new approach is presented to facilitate research in the state estimation of linear systems using continuous orthogonal functions. The principle of the Luenberger observer is utilized in developing simple algebraic expressions for the estimates of states. This approach has the distinct advantage that the smoothing effect of integration reduces the effect of noise. Hence, this observer gives acceptable estimates of the states in the presence of zero-mean observation noise, even without a filter. Results of simulation indicate that the proposed method works quite well. In addition, the algorithms are recursive and suitable for on-line implementation.

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.

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.

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 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).

Kalman filter estimation of electrical machine parameters

Abstract: The authors have reported on the computer implementation of a Kalman filter for the determination of system parameters. This paper describes the application of the technique in the determination of the parameters of a d.c. motor. The validity of the technique is confirmed by comparing the experimentally observed machine response with the theoretical response predicted from the estimated parameters.

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.

Function Observer Design for a Class of Time Varying Systems

Abstract: This paper presents, for the first time, a reduced order function observer design algorithm for uniformly observable time varying systems. The resulting observer will have a constant Jordan form system matrix with arbitrarily given poles. The order reduction of the function observer is also clarified and clearly explained.

Design of a reduced-order adaptive observer

Abstract: A reduced-order implicit adaptive observer for a single-input single-output time-invariant nth-order linear system is considered. A state vector described in observable canonical form is explicitly parameterised in terms of the unknown system parameters and the filtered input and output vectors. For parameter estimation, a least-square-type adaptation scheme that can afford an arbitrarily fast exponential convergence is proposed.

A Theoretical Analysis of the Round-Off Propagation in Different Kalman Filter Implementations

Abstract: A theoretical analysis is made of the error propagation due to numerical round-off 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. From these error models, new insights about the applicability of the different filters and their sensitivity to round-off, is obtained. It is shown that the CKF may become completely unreliable when the original plant is unstable, and that this is easily circumvented by a number of techniques. The square root filters, often quoted to possess a conditioning or sensitivity that is the square root of that of the CKF, are shown to possess this property only for the computation of the covariance of the filtered signal and not for the computation of the Kalman gain or the filtered estimate. Finally, the Chandrasekhar filter is shown to be numerically unstable.