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DOE OSTI2021

Penalized ensemble Kalman filters for high dimensional non-linear systems

The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of models, updated with data, to track the time evolution of a usually non-linear system. It does so by using an empirical approximation to the well-known Kalman filter. However, its performance can suffer when the ensemble size is smaller than the state space, as is often necessary for computationally burdensome models. This scenario means that the empirical estimate of the state covariance is not full rank and possibly quite noisy. To solve this problem in this high dimensional regime, we propose a computationally fast and easy to implement algorithm called the penalized ensemble Kalman filter (PEnKF). Under certain conditions, it can be theoretically proven that the PEnKF will be accurate (the estimation error will converge to zero) despite having fewer ensemble members than state dimensions. Further, as contrasted to localization methods, the proposed approach learns the covariance structure associated with the dynamical system. These theoretical results are supported with simulations of several non-linear and high dimensional systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

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.↗

NASA NTRS1997

Implicit Kalman filtering

For an implicitly defined discrete system, a new algorithm for Kalman filtering is developed and an efficient numerical implementation scheme is proposed. Unlike the traditional explicit approach, the implicit filter can be readily applied to ill-conditioned systems and allows for generalization to descriptor systems. The implementation of the implicit filter depends on the solution of the congruence matrix equation (A1)(Px)(AT1) = Py. We develop a general iterative method for the solution of this equation, and prove necessary and sufficient conditions for convergence. It is shown that when the system matrices of an implicit system are sparse, the implicit Kalman filter requires significantly less computer time and storage to implement as compared to the traditional explicit Kalman filter. Simulation results are presented to illustrate and substantiate the theoretical developments.

Non-NASA Center↗

NASA NTRS1979

The discrete-time compensated Kalman filter

A suboptimal dynamic compensator to be used in conjunction with the ordinary discrete-time Kalman filter is derived. The resultant compensated Kalman filter has the property that steady-state bias estimation errors, resulting from modelling errors, are eliminated. The implementation of the compensated Kalman filter involves the use of accumulators in the residual channels in addition to the nominal dynamic model of the stochastic system.

Lee, W.-H.↗

NASA NTRS1984

Kalman filter modeling

The formulation of appropriate state-space models for Kalman filtering applications is studied. The so-called model is completely specified by four matrix parameters and the initial conditions of the recursive equations. Once these are determined, the die is cast, and the way in which the measurements are weighted is determined foreverafter. Thus, finding a model that fits the physical situation at hand is all important. Also, it is often the most difficult aspect of designing a Kalman filter. Formulation of discrete state models from the spectral density and ARMA random process descriptions is discussed. Finally, it is pointed out that many common processes encountered in applied work (such as band-limited white noise) simply do not lend themselves very well to Kalman filter modeling.

Brown, R. G.↗

NASA NTRS2007

Zero Gyro Kalman Filtering in the Presence of a Reaction Wheel Failure

Typical implementation of Kalman filters for spacecraft attitude estimation involves the use of gyros for three-axis rate measurements. When there are less than three axes of information available, the accuracy of the Kalman filter depends highly on the accuracy of the dynamics model. This is particularly significant during the transient period when a reaction wheel with a high momentum fails, is taken off-line, and spins down. This paper looks at how a reaction wheel failure can affect the zero-gyro Kalman filter performance for the Hubble Space Telescope and what steps are taken to minimize its impact.

Hur-Diaz, Sun↗

NASA NTRS2007

Zero Gyro Kalman Filtering in the presence of a Reaction Wheel Failure

Hur-Diaz, Sun↗

NASA NTRS1991

The History Of The Kalman Filter

Paper presents historical view of adaptation of Kalman filtering techniques to aerospace applications and eventually to fields as diverse as exploration for oil and control of powerplants. Describes scientific breakthroughs and reformulations that transformed Kalman filtering techniques into fundamental tool for analyzing and solving broad class of estimation problems.

Mcgee, Leonard A.↗

NASA NTRS2000

Application of a Reduced Order Kalman Filter to Initialize a Coupled Atmosphere-Ocean Model: Impact on the Prediction of El Nino

A reduced order Kalman Filter, based on a simplification of the Singular Evolutive Extended Kalman (SEEK) filter equations, is used to assimilate observed fields of the surface wind stress, sea surface temperature and sea level into the nonlinear coupled ocean-atmosphere model. The SEEK filter projects the Kalman Filter equations onto a subspace defined by the eigenvalue decomposition of the error forecast matrix, allowing its application to high dimensional systems. The Zebiak and Cane model couples a linear reduced gravity ocean model with a single vertical mode atmospheric model of Zebiak. The compatibility between the simplified physics of the model and each observed variable is studied separately and together. The results show the ability of the model to represent the simultaneous value of the wind stress, SST and sea level, when the fields are limited to the latitude band 10 deg S - 10 deg N. In this first application of the Kalman Filter to a coupled ocean-atmosphere prediction model, the sea level fields are assimilated in terms of the Kelvin and Rossby modes of the thermocline depth anomaly. An estimation of the error of these modes is derived from the projection of an estimation of the sea level error over such modes. This method gives a value of 12 for the error of the Kelvin amplitude, and 6 m of error for the Rossby component of the thermocline depth. The ability of the method to reconstruct the state of the equatorial Pacific and predict its time evolution is demonstrated. The method is shown to be quite robust for predictions I up to six months, and able to predict the onset of the 1997 warm event fifteen months before its occurrence.

Ballabrera-Poy, Joaquim↗

NASA NTRS2000

Application of a Reduced Order Kalman Filter to Initialize a Coupled Atmosphere-Ocean Model: Impact on the Prediction of El Nino

A reduced order Kalman Filter, based on a simplification of the Singular Evolutive Extended Kalman (SEEK) filter equations, is used to assimilate observed fields of the surface wind stress, sea surface temperature and sea level into the nonlinear coupled ocean-atmosphere model of Zebiak and Cane. The SEEK filter projects the Kalman Filter equations onto a subspace defined by the eigenvalue decomposition of the error forecast matrix, allowing its application to high dimensional systems. The Zebiak and Cane model couples a linear reduced gravity ocean model with a single vertical mode atmospheric model of Zebiak. The compatibility between the simplified physics of the model and each observed variable is studied separately and together. The results show the ability of the model to represent the simultaneous value of the wind stress, SST and sea level, when the fields are limited to the latitude band 10 deg S - 10 deg N In this first application of the Kalman Filter to a coupled ocean-atmosphere prediction model, the sea level fields are assimilated in terms of the Kelvin and Rossby modes of the thermocline depth anomaly. An estimation of the error of these modes is derived from the projection of an estimation of the sea level error over such modes. This method gives a value of 12 for the error of the Kelvin amplitude, and 6 m of error for the Rossby component of the thermocline depth. The ability of the method to reconstruct the state of the equatorial Pacific and predict its time evolution is demonstrated. The method is shown to be quite robust for predictions up to six months, and able to predict the onset of the 1997 warm event fifteen months before its occurrence.

Ballabrera-Poy, J.↗

NASA NTRS1988

An analysis of the Kalman filter in the Gamma Ray Observatory (GRO) onboard attitude determination subsystem

The Gamma Ray Observatory (GRO) spacecraft needs a highly accurate attitude knowledge to achieve its mission objectives. Utilizing the fixed-head star trackers (FHSTs) for observations and gyroscopes for attitude propagation, the discrete Kalman Filter processes the attitude data to obtain an onboard accuracy of 86 arc seconds (3 sigma). A combination of linear analysis and simulations using the GRO Software Simulator (GROSS) are employed to investigate the Kalman filter for stability and the effects of corrupted observations (misalignment, noise), incomplete dynamic modeling, and nonlinear errors on Kalman filter. In the simulations, on-board attitude is compared with true attitude, the sensitivity of attitude error to model errors is graphed, and a statistical analysis is performed on the residuals of the Kalman Filter. In this paper, the modeling and sensor errors that degrade the Kalman filter solution beyond mission requirements are studied, and methods are offered to identify the source of these errors.

Snow, Frank↗

NASA NTRS1976

A numerical comparison of discrete Kalman filtering algorithms - An orbit determination case study

An improved Kalman filter algorithm based on a modified Givens matrix triangularization technique is proposed for solving a nonstationary discrete-time linear filtering problem. The proposed U-D covariance factorization filter uses orthogonal transformation technique; measurement and time updating of the U-D factors involve separate application of Gentleman's fast square-root-free Givens rotations. Numerical stability and accuracy of the algorithm are compared with those of the conventional and stabilized Kalman filters and the Potter-Schmidt square-root filter, by applying these techniques to a realistic planetary navigation problem (orbit determination for the Saturn approach phase of the Mariner Jupiter-Saturn Mission, 1977). The new algorithm is shown to combine the numerical precision of square root filtering with the efficiency of the original Kalman algorithm.

Thornton, C. L.↗

NASA NTRS2020

Analytical Redundancy Using Kalman Filters for Rocket Engine Sensor Validation

The use of sensor redundancy is crucial in aerospace systems to maintain safe, reliable operation. While hardware redundancy is more common in application, analytical redundancy can provide a viable alternative in systems where the installation of multiple redundant sensors is not viable. To this end, the use of Kalman filters to analytically validate sensor measurements within rocket engines was explored. First, a dynamic model of the RS 25 engine, a derivative of the Space Shuttle Main Engine (SSME), was reduced to a subset of relations, focused around the main combustion chamber pressure. These relations were used within the Kalman filter algorithm to generate an estimate of sensor measurements to be compared with true measurements for data validation purposes. By using a bank of Kalman filters, the residuals between the estimated and true measurements were used to detect and isolate sensor faults. Through fault simulations, the sensor validation performance of this Kalman filter bank design was compared to a hardware redundancy check. Sensor bias and drift faults of various magnitudes were injected into nominal RS 25 engine test data. Results for both approaches show comparable fault detection with most bias faults found nearly instantaneously by both algorithms. Drift fault detection results show certain cases where one algorithm is faster than the other. The key advantage of the Kalman filter algorithm is shown in fault isolation performance where it can isolate faults between two redundant sensors while the hardware redundancy comparisons cannot.

sensors↗

NASA NTRS1985

Discovery of the Kalman filter as a practical tool for aerospace and industry

The sequence of events which led the researchers at Ames Research Center to the early discovery of the Kalman filter shortly after its introduction into the literature is recounted. The scientific breakthroughs and reformulations that were necessary to transform Kalman's work into a useful tool for a specific aerospace application are described. The resulting extended Kalman filter, as it is now known, is often still referred to simply as the Kalman filter. As the filter's use gained in popularity in the scientific community, the problems of implementation on small spaceborne and airborne computers led to a square-root formulation of the filter to overcome numerical difficulties associated with computer word length. The work that led to this new formulation is also discussed, including the first airborne computer implementation and flight test. Since then the applications of the extended and square-root formulations of the Kalman filter have grown rapidly throughout the aerospace industry.

Mcgee, L. A.↗

NASA NTRS1991

Mass property identification - A comparison study between extended Kalman filter and neuro-filter approaches

Two algorithms, extended Kalman filter and neuro-filter, are formulated to perform mass property identification for the Space Station Freedom. Control moment gyros that are part of the Station's basic momentum management system are chosen to provide input excitation in the form of applied torques. These torques together with the measured angular body rate responses are supplied to the filters. From these data, both algorithms are shown to accurately identify the station mass properties when excitation levels are high and balanced between axes. The neuro-filter, however, is shown to be more robust and to perform well even with weakly persistent, unbalanced signals contaminated with noise.

Lam, Quang↗

NASA NTRS1993

Estimation of Kalman filter gain from output residuals

This paper presents a procedure to estimate the Kalman filter gain from input-output measurement data with a given system model. The system model can be a finite element model or an experimental model from any identification method. The procedure consists of three basic steps. First, the stochastic portion related to the residuals of the response is computed. Second, the coefficients of a linear difference model for the stochastic portion are estimated by a least-squares solution that minimizes the filter residual. Third, the Kalman filter gain is computed from these model coefficients. Experimental results are presented to illustrate the usefulness of the developed procedure.

Juang, Jer-Nan↗