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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

New insights in the numerical reliability properties of existing Kalman filter implementations

The convergence properties of Kalman filter algorithms are investigated analytically. A theoretical error analysis is performed on four types of algorithms, as defined by Anderson and Moore (1979): (1) a conventional Kalman filter, (2) a square-root (SR) covariance filter, (3) the Chandrasekhar SR filter, and (4) an SR information filter. The derivations are given in detail, and numerical results for the flight-path reconstruction problem studied by Verhaegen (1987) are presented in tables and graphs. It is shown that error propagation in algorithms (1) and (2) is sensitive to the condition number of the innovation-signal covariance matrix and the spectral norm of the filter state-transition matrix, whereas other parameters are dominant in (3) and (4). Filter (2) is found to be the most reliable for the class of problems studied.

Verhaegen, M. H.↗
NASA NTRS1978

Kalman filter divergence and aircraft motion estimators

Kalman filters designed for many aerospace systems turn out to be unsatisfactory. The estimate errors become large compared to the errors predicted by the theory ('divergence'). One of the principal causes of this failure is that the system model contains states or modes that are undisturbed by the modeled process noise, and are neutrally stable (NS). One cure for such problems is periodic restarting of a time-varying Kalman filter. Other cures include minimum variance observers with eigenvalue constraints, added noise, pole-shifting, and destabilization. Several examples are given, including effective time-invariant estimators for the longitudinal and lateral motions of an airplane where several NS modes are undisturbed by wind gusts. An interpretation of these estimators as a 'strapdown IMU' without accelerometers, gimbaled gyros, or servos is given.

Bryson, A. E., Jr.↗
NASA NTRS2024

Iterated Sigma-Point Kalman Filtering for Trajectory Reconstruction

Two second-order Kalman filtering methods are added to the New Statistical Trajec-tory Estimation Program (NewSTEP). The two filters are of the class of Sigma-Point Kalman Filters, namely the Unscented Kalman Filter and the Divided Difference Filter. Both filters are implemented with an iterative measurement update step. Results from simulation test case and a flight data test case are provided to show the filter performance. The results indicate that the new second-order filters pro-duce trajectory estimates with reduced uncertainties when compared to the existing first-order filtering method.

Sophia Vedvik↗
NASA NTRS2003

An optimal modification of a Kalman filter for time scales

The Kalman filter in question, which was implemented in the time scale algorithm TA(NIST), produces time scales with poor short-term stability. A simple modification of the error covariance matrix allows the filter to produce time scales with good stability at all averaging times, as verified by simulations of clock ensembles.

time scale Kalman filter↗
NASA NTRS1995

A Reduced Dimension Static, Linearized Kalman Filter and Smoother

An approximate Kalman filter and smoother, based on approximations of the state estimation error covariance matrix, is described. Approximations include a reduction of the effective state dimension, use of a static asymptotic error limit, and a time-invariant linearization of the dynamic model for error integration. The approximations lead to dramatic computational savings in applying estimation theory to large complex systems. Examples of use come from TOPEX/POSEIDON.

Kalman filter smoother modeling ocean simulation↗
NASA NTRS1991

Estimation of optimal Kalman filter gain from non-optimal filter residuals

This paper presents a novel method of estimating the optimal steady state Kalman filter gain of a linear discrete time-invariant system from a non-optimal Kalman filter residual sequence. The relation between the optimal residual sequence and a signal derived from the non-optimal residual sequence is described by a Moving Average (MA) model whose coefficients are expressed in terms of the state space parameters and the optimal steady state Kalman filter gain. In order to identify the MA model, a whitening filter of the derived signal, which corresponds to an AutoRegressive (AR) model of the signal, is first identified using the least-squares method. Then the inverse filter of the whitening filter, which corresponds to the MA model, is calculated. From the coefficients of the identified MA model, the optimal steady state Kalman filter gain can be obtained. Numerical example is provided to illustrate the feasibility of this approach.

Chen, Chung-Wen↗
NASA NTRS1990

Geodesy by radio interferometry - The application of Kalman filtering to the analysis of very long baseline interferometry data

The application of Kalman filtering techniques to the analysis of VLBI data is discussed. The VLBI observables are geometrically related to the geodetic and astrometric parameters which can be determined from them. However, contributions to the observables from the clocks at, and the atmospheres above, the VLBI sites must be accounted for if reliable estimates of geodetic and astrometric parameters are to be obtained. Here, an implementation of a Kalman filter to account for stochastic behavior on those parameters which vary during the course of a VLBI experiment is discussed. Both the nature of the stochastic processes which should be used in the model for the VLBI data and the implementation of the Kalman filter estimator are considered. It is concluded that the Kalman filter is appropriate for analyzing VLBI data.

Herring, Thomas A.↗
NASA NTRS1987

Improved understanding of the loss-of-symmetry phenomenon in the conventional Kalman filter

This paper corrects an unclear treatment of the conventional Kalman filter implementation as presented by M. H. Verhaegen and P. van Dooren in Numerical aspects of different Kalman filter implementations, IEEE Trans. Automat. Contr., v. AC-31, no. 10, pp. 907-917, 1986. It is shown that habitual, incorrect implementation of the Kalman filter has been the major cause of its sensitivity to the so-called loss-of-symmetry phenomenon.

Verhaegen, M. H.↗
NASA NTRS1989

Improved understanding of the loss-of-symmetry phenomenon in the conventional Kalman filter

This paper corrects an unclear statement of the conventional Kalman filter implementation as presented by M. H. Verhaegen and P. van Dooren in Numerical aspects of different Kalman filter implementations, IEEE Trans. Automat. Contr., v. AC-31, no. 10, pp. 907-917, 1986. It is shown that the habitual, incorrect implementation of the Kalman filter has been the major cause of its insensitivity to the so-called loss-of-symmetry phenomenon.

Verhaegen, M. H.↗
NASA NTRS1988

Relationships of earthquakes (and earthquake-associated mass movements) and polar motion as determined by Kalman filtered, Very-Long-Baseline-Interferometry

A Kalman filter was designed to yield optimal estimates of geophysical parameters from Very Long Baseline Interferometry (VLBI) group delay data. The geophysical parameters are the polar motion components, adjustments to nutation in obliquity and longitude, and a change in the length of day parameter. The VLBI clock (and clock rate) parameters and atmospheric zenith delay parameters are estimated simultaneously. Filter background is explained. The IRIS (International Radio Interferometric Surveying) VLBI data are Kalman filtered. The resulting polar motion estimates are examined. There are polar motion signatures at the times of three large earthquakes occurring in 1984 to 1986: Mexico, 19 September, 1985 (Magnitude M sub s = 8.1); Chile, 3 March, 1985 (M sub s = 7.8); and Taiwan, 14 November, 1986 (M sub s = 7.8). Breaks in polar motion occurring about 20 days after the earthquakes appear to correlate well with the onset of increased regional seismic activity and a return to more normal seismicity (respectively). While the contribution of these three earthquakes to polar motion excitations is small, the cumulative excitation due to earthquakes, or seismic phenomena over a Chandler wobble damping period may be significant. Mechanisms for polar motion excitation due to solid earth phenomena are examined. Excitation functions are computed, but the data spans are too short to draw conclusions based on these data.

Preisig, Joseph Richard Mark↗
NASA NTRS2004

Evaluation of an Enhanced Bank of Kalman Filters for In-Flight Aircraft Engine Sensor Fault Diagnostics

In this paper, an approach for in-flight fault detection and isolation (FDI) of aircraft engine sensors based on a bank of Kalman filters is developed. This approach utilizes multiple Kalman filters, each of which is designed based on a specific fault hypothesis. When the propulsion system experiences a fault, only one Kalman filter with the correct hypothesis is able to maintain the nominal estimation performance. Based on this knowledge, the isolation of faults is achieved. Since the propulsion system may experience component and actuator faults as well, a sensor FDI system must be robust in terms of avoiding misclassifications of any anomalies. The proposed approach utilizes a bank of (m+1) Kalman filters where m is the number of sensors being monitored. One Kalman filter is used for the detection of component and actuator faults while each of the other m filters detects a fault in a specific sensor. With this setup, the overall robustness of the sensor FDI system to anomalies is enhanced. Moreover, numerous component fault events can be accounted for by the FDI system. The sensor FDI system is applied to a commercial aircraft engine simulation, and its performance is evaluated at multiple power settings at a cruise operating point using various fault scenarios.

Kobayashi, Takahisa↗
NASA NTRS2006

Real-Time Diagnosis of Faults Using a Bank of Kalman Filters

A new robust method of automated real-time diagnosis of faults in an aircraft engine or a similar complex system involves the use of a bank of Kalman filters. In order to be highly reliable, a diagnostic system must be designed to account for the numerous failure conditions that an aircraft engine may encounter in operation. The method achieves this objective though the utilization of multiple Kalman filters, each of which is uniquely designed based on a specific failure hypothesis. A fault-detection-and-isolation (FDI) system, developed based on this method, is able to isolate faults in sensors and actuators while detecting component faults (abrupt degradation in engine component performance). By affording a capability for real-time identification of minor faults before they grow into major ones, the method promises to enhance safety and reduce operating costs. The robustness of this method is further enhanced by incorporating information regarding the aging condition of an engine. In general, real-time fault diagnostic methods use the nominal performance of a "healthy" new engine as a reference condition in the diagnostic process. Such an approach does not account for gradual changes in performance associated with aging of an otherwise healthy engine. By incorporating information on gradual, aging-related changes, the new method makes it possible to retain at least some of the sensitivity and accuracy needed to detect incipient faults while preventing false alarms that could result from erroneous interpretation of symptoms of aging as symptoms of failures. The figure schematically depicts an FDI system according to the new method. The FDI system is integrated with an engine, from which it accepts two sets of input signals: sensor readings and actuator commands. Two main parts of the FDI system are a bank of Kalman filters and a subsystem that implements FDI decision rules. Each Kalman filter is designed to detect a specific sensor or actuator fault. When a sensor or actuator fault occurs, large estimation errors are generated by all filters except the one using the correct hypothesis. By monitoring the residual output of each filter, the specific fault that has occurred can be detected and isolated on the basis of the decision rules. A set of parameters that indicate the performance of the engine components is estimated by the "correct" Kalman filter for use in detecting component faults. To reduce the loss of diagnostic accuracy and sensitivity in the face of aging, the FDI system accepts information from a steady-state-condition-monitoring system. This information is used to update the Kalman filters and a data bank of trim values representative of the current aging condition.

Kobayashi, Takahisa↗
NASA NTRS1993

Multirate and event-driven Kalman filters for helicopter flight

A vision-based obstacle detection system that provides information about objects as a function of azimuth and elevation is discussed. The range map is computed using a sequence of images from a passive sensor, and an extended Kalman filter is used to estimate range to obstacles. The magnitude of the optical flow that provides measurements for each Kalman filter varies significantly over the image depending on the helicopter motion and object location. In a standard Kalman filter, the measurement update takes place at fixed intervals. It may be necessary to use a different measurement update rate in different parts of the image in order to maintain the same signal to noise ratio in the optical flow calculations. A range estimation scheme that accepts the measurement only under certain conditions is presented. The estimation results from the standard Kalman filter are compared with results from a multirate Kalman filter and an event-driven Kalman filter for a sequence of helicopter flight images.

Sridhar, Banavar↗
NASA NTRS2021

Multi-Pass Sequential Mini-Batch Stochastic Gradient Descent Algorithms for Noise Covariance Estimation in Adaptive Kalman Filtering

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of applications. Although this problem has a long history, reliable algorithms for their estimation were scant, and necessary and sufficient conditions for identifiability of the covariances were in dispute until recently. Necessary and sufficient conditions for covariance estimation and a batch estimation algorithm were presented in our previous study. This paper presents stochastic gradient descent algorithms for noise covariance estimation in adaptive Kalman filters that are an order of magnitude faster than the batch method for similar or better root mean square error. More significantly, these algorithms are applicable to non-stationary systems where the noise covariances can occasionally jump up or down by an unknown magnitude. The computational efficiency of the new algorithms stems from adaptive thresholds for convergence, recursive fading memory estimation of the sample cross-correlations of the innovations, and accelerated stochastic gradient descent algorithms. The comparative evaluation of the proposed methods on a number of test cases demonstrates their computational efficiency and accuracy.

Adaptive Kalman filtering↗
NASA NTRS1993

Kalman-Filter Estimation Of Unmeasured Output Variables

Kalman-filter "observer" algorithm used in estimating unmeasured output variables of turbofan engine. Unmeasured output variables needed along with measured output variables as feedback data to characterize state of aircraft-and-engine system for control purposes. Subsystem or processor implementing Kalman-filter algorithm incorporated into adaptive digital control system, called "performance-seeking" control system, developed to compute engine-control settings to optimize performance of aircraft under given pseudo-steady-state cruise conditions. Basic concept of algorithm not restricted to aircraft-engine application and can be extended to internal-combustion engines in land vehicles.

Gilyard, Glenn B.↗
NASA NTRS2014

Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms

This paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate non-negativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. In two examples, an update that includes a non-negativity constraint is able to properly describe the transport of a sharp feature (e.g., a triangle or cone). A number of implementation questions still need to be addressed, particularly the need to develop a computationally efficient quadratic programming update for large ensemble.

EnKF↗
NASA NTRS1986

Systolic VLSI for Kalman filters

A novel two-dimensional parallel computing method for real-time Kalman filtering is presented. The mathematical formulation of a Kalman filter algorithm is rearranged to be the type of Faddeev algorithm for generalizing signal processing. The data flow mapping from the Faddeev algorithm to a two-dimensional concurrent computing structure is developed. The architecture of the resulting processor cells is regular, simple, expandable, and therefore naturally suitable for VLSI chip implementation. The computing methodology and the two-dimensional systolic arrays are useful for Kalman filter applications as well as other matrix/vector based algebraic computations.

Yeh, H.-G.↗