Kalman Filter Theory Research Articles

REAL-TIME PREDICTION ALGORITHM WITH A DISTRIBUTED HYDROLOGICAL MODEL USING KALMAN FILTER

The Kalman filter theory is coupled with a distributed hydrological model to update spatially distributed state variables by using several techniques proposed here. To acquire the total water storages of a basin from discharge observations at the outlet, a Q-S curve is used as an observation equation. After updating total water storage with the Kalman filter, the ratio method is introduced to reset the distributed storage amount of a basin, maintaining the spatially distributed pattern. A Monte Carlo simulation is adopted to predict state variables and error variance propagations. A distributed model coupled with the Kalman filter theory gives updated simulation results with improved forecasting accuracy.

A Heuristic Kalman Filter for a Class of Nonlinear Systems

One of the basic assumptions involved in the optimality of the Kalman filter theory is that the system under consideration must be linear. If the model is nonlinear, a linearization procedure is usually performed in deriving the filtering equations. This approach requires the nonlinear system dynamics to be differentiable. This note is an attempt to develop a heuristic Kalman filter for a class of nonlinear systems, with bounded first-order growth, that does not require the system dynamics to be differentiable. The proposed filter approximates the nonlinear state function by its state argument multiplied by a particular gain matrix only in the recursion of the estimation error covariance matrix. Under certain conditions, the error covariance remains bounded by bounds which can be precomputed from noise and system models, and the upper bound tends to zero when the state noise covariance tends to zero. A numerical example, with backlash nonlinearity, is also added.

Estimation of Backlash in Automotive Powertrains — An Experimental Validation

In automotive powertrains, backlash imposes well-known limitations on the quality of control and hence the driveability. In order to increase the controller performance, knowledge of the backlash properties (size and current position) is needed. In previous papers, nonlinear estimators for backlash size and position are described, based on Kalman filtering theory. The size and position estimators are combined, and the result is an estimate of the state of the backlash. In this paper, experimental results from vehicle tests show that the estimates are of high quality, and hence useful for improving backlash compensation functions in the powertrain control system.

Realizable Linear and Decision Feedback Equalizers: Properties and Connections

Recently, there has been renewed interest in the use of infinite impulse response (IIR) linear equalizers (LEs) for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures (DFEs). Theoretical justification for such an approach is usually given assuming unconstrained filters, which are not causal and therefore not implementable in practice. We present an analysis of realizable (i.e., causal, stable, and of finite degree) minimum mean square error (MMSE) equalizers for single-input multiple-output channels, both in the LE and DFE cases, focusing on their structures and filter orders, as well as the connections between them. The DFE resulting from rearranging the MMSE LE within a decision feedback loop is given special attention. It is shown that although this DFE does not in general coincide with the MMSE DFE, it still enjoys certain optimality conditions. The main tools employed are the Wiener theory of minimum variance estimation and Kalman filtering theory, which show interesting properties of the MMSE equalizers not revealed by previous polynomial approaches.

Multi-sensor optimal information fusion Kalman filters with applications

Using Kalman filtering theory, a new multi-sensor optimal information fusion algorithm weighted by matrices is presented in the linear minimum variance sense, which is equivalent to the maximum likelihood fusion algorithm under the assumption of normal distributions. The algorithm considers the correlation among local estimation errors, and it involves the inverse of certain matrix with high dimension. Another two new multi-sensor suboptimal information fusion algorithms weighted by vectors and weighted by scalars are given for reducing the computational burden and increasing the real-time property. Based on these fusion algorithms, the multi-sensor optimal and suboptimal information fusion Kalman filters with two-layer fusion structures are given. The simulation researches of the comparisons among them as well as the centralized filter in a radar tracking system with three sensors show their effectiveness.

Optimal estimation of multicast membership

This paper addresses optimal online estimation of the size of a multicast group. Three distinct approaches are used. The first one builds on Kalman filter theory to derive the MSE-optimal estimator in a heavy-traffic regime. Under more general assumptions, the second approach uses Wiener filter theory to compute the MSE-optimal linear filter. The third approach develops the best first-order linear filter from which an estimator that holds for any on-time distribution is derived. Our estimators are tested on real video traces and exhibit good performance. The paper also provides guidelines on how to tune the parameters involved in the schemes in order to achieve high-quality estimation while simultaneously avoiding feedback implosion.

Kalman filtering for fuzzy discrete time dynamic systems

Abstract This paper uses Kalman filter theory to design a state estimator for noisy discrete time Takagi–Sugeno (T–S) fuzzy models. One local filter is designed for each local linear model using standard Kalman filter theory. Steady state solutions can be found for each of the local filters. Then a linear combination of the local filters is used to derive a global filter. The local filters are time-invariant, which greatly reduces the computational complexity of the global filter. The global filter is shown to be unbiased and (under certain conditions) stable. In addition, under the approximation of uncorrelatedness among the local models, the global filter is shown to be minimum variance. The proposed state estimator is demonstrated on a vehicle tracking problem and a backing up truck–trailer example.

Constrained numerical optimization of PCR/PLSR predictors

Assuming a fully known latent variables (LV) model, the optimal multivariate calibration predictor is found from Kalman filtering theory. From this follows the best possible column space for a loading weight matrix W opt. in a predictor based on the latent variables, and thus the optimal factorization of the regressor matrix X. Although the optimal predictor cannot be directly determined in a practical case, we may still make an attempt to find it. The paper presents a simple algorithm for a constrained numerical search for a W opt. matrix spanning the optimal column space, using a principal component analysis (PCR) or a partial least squares (PLS) factorization as a starting point. The constraint is necessary in order to avoid overfitting, and it is based on an assumption of a smooth predictor. A simulation example and data from a metal ion mixture experiment are used to demonstrate the feasibility of the proposed method.

A Practical Digital Signal Carrier Acquisition based on Kalman Filter Theory

Traditionally Extended Kalman Filter has been proposed and studied for the acquisition of a digital carrier signal. However, this technique imposes a heavy load on the processor. And high performances cannot be expected for the poor linearity, in most cases. To overcome these problems, this paper shows that one can use a simple, purely linear Kalman Filter which consists of two states variables - phase and frequency. This new technique selects the carrier phase as the input to the filter, instead of a pair of orthogonal signal amplitudes. The filtering logic is made up of only 4 additions and 2 multiplications. The results of both simulations and experiments show that this filter can acquire the carrier signal within 10 symbols with a probability of 98 % during the initial phase, even when the frequency offset is as large as 20 % of the symbol rate frequency at C/N=6dB. In the steady state, the measured BER is close to the theoretical values. While delivering a similar performance mentioned above, this filter can operate even when the carrier frequency deviates from the expected figure.

An extension of a data assimilation method based on the application of the Fokker–Planck equation

A data assimilation method based on the Kalman filter theory and on the Fokker–Planck equation is extended to assimilate Atlantic Ocean data into a new version of the well-known Modular Ocean Model (MOM_3) from NOAA/GFDL. This extension enables assimilation of non-uniformly distributed data in space and time. Numerical experiments with Levitus atlas data are carried out with the ocean model configured at a low resolution. Some results of these experiments as well as other possible expansions are discussed.

Assimilation of sea surface salinity in a tropical Oceanic General Circulation Model (OGCM): A twin experiment approach

Observing ocean surface salinity in the global ocean is a challenging issue for future years' oceanographic activities. It is motivated by the active role of salinity that is now well recognized in ocean dynamics and ocean/atmosphere exchanges. This is particularly evident in the case of the El Niño Southern Oscillation (ENSO) phenomenon in the tropical Pacific Ocean. Improvements to ocean state estimation and predictions will require that salinity observations must be taken into proper account in conjunction with temperature and altimetric data. The sensitivity of a primitive equation model of the tropical Pacific Ocean to sea surface salinity (SSS) is studied through the use of a data assimilation technique in the rather academic “twin experiment” context. The data assimilation technique used, the Singular Evolutive Extended Kalman (SEEK) filter, is derived from the conventional Kalman filter theory. The paper explains why such a sophisticated technique is necessary. Indeed, an empirical scheme such as the Newtonian relaxation method, used in the same conditions, fails to constrain either the observed (surface) variable or the other components of the state vector. Within the experimental context chosen, the assimilation of SSS data with the SEEK filter is able to constrain most of the model variables linked with the SSS signal. SSS information, in particular, appears relatively successful in restoring zonal velocity, which is an important variable in warm/fresh pool migration, and in simulating a barrier layer in the atmospheric convergence zones. The final analysis errors are small and stable over time. This is widely true when simulating satellite SSS observations based on the GODAE criteria (0.2 psu error, 200 km, 10 days), which shows the potential of these observations. To extend these results to a real context, the problems of model‐data bias and unknown error covariances must be addressed as they are actually a strong limitation in assimilation performance when assimilating any real data set.

PCR/PLSR optimization based on noise covariance estimation and Kalman filtering theory

AbstractThe theoretical connection between principal component regression (PCR) and partial least squares regression (PLSR) on one hand and Kalman filtering (KF) on the other is known from earlier work. In the present paper we investigate the possibilities to use latent variables modeling and KF theory as means for optimization of ordinary PLSR and PCR predictors, based on the prerequisite of prior X noise covariance estimates facilitated e.g. by more X than y observations. The result is a new PLSR optimization method, while the PCR optimization turns out to be identical with an earlier known method. A simulation example and two real‐world data examples supporting the theoretical development are presented. The treatment is limited to cases with only one response variable, although an extension to multiresponse cases is also possible. Copyright © 2002 John Wiley & Sons, Ltd.

On the oceanic impact of a data-assimilation method in a coupled ocean-land-atmosphere model

Based on the Kalman filter theory, a new data-assimilation method has been used to improve the 3-D oceanic temperature field of the Center for Ocean-Land-Atmosphere Studies (COLA) coupled general circulation model. This method is applied to assimilate surface and subsurface temperature of in situ measurements from the Pilot Research Moored Array in the Tropical Atlantic project (PIRATA). The assimilation of the PIRATA data produces an improved representation of the thermal state of the ocean and allows a better estimation of other oceanographic quantities, like meridional heat fluxes and zonal currents. The present paper focuses on the tropical Atlantic and, in particular, it contains new reconstructed temperature profiles. One-month forecast experiments during 1999 were performed and the impact of the assimilation is discussed.

Using Improved Background-Error Covariances from an Ensemble Kalman Filter for Adaptive Observations

A method for determining adaptive observation locations is demonstrated. This method is based on optimal estimation (Kalman filter) theory; it determines the observation location that will maximize the expected improvement, which can be measured in terms of the expected reduction in analysis or forecast variance. This technique requires an accurate model for background error statistics that vary both in space and in time. Here, these covariances are generated using an ensemble Kalman filter assimilation scheme. A variant is also developed that can estimate the analysis improvement in data assimilation schemes where background error statistics are less accurate. This approach is demonstrated using a quasigeostrophic channel model under perfect-model assumptions. The algorithm is applied here to find the supplemental rawinsonde location to add to a regular network of rawinsondes that will reduce analysis errors the most. The observation network is configured in this experiment so there is a data void in the western third of the domain. One-hundred-member ensembles from three data assimilation schemes are tested as input to the target selection procedure, two variants of the standard ensemble Kalman filter and a third perturbed observation (3DVAR) ensemble. The algorithm is shown to find large differences in the expected variance reduction depending on the observation location, the flow of the day, and the ensemble used in the adaptive observation algorithm. When using the two variants of the ensemble Kalman filter, the algorithm defined consistently similar adaptive locations to each other, and assimilation of the adaptive observation typically reduced analysis errors significantly. When the 3DVAR ensemble was used, the algorithm picked very different observation locations and the analyses were not improved as much. The amount of improvement from assimilating a supplemental adaptive observation instead of a fixed observation in the middle of the void depended on whether the observation was assimilated sporadically or during every analysis cycle. For sporadic assimilation, the adaptive observation provided a dramatic improvement relative to the supplemental fixed observation. When an adaptive observation was regularly assimilated every cycle, the improvement was smaller. For the sporadic assimilation of an adaptive observation, targeting based simply on the maximum spread in background forecasts provided similar target locations and similar analysis improvements to those generated with the full algorithm. The improvement from the regular assimilation of an adaptive observation based on the spread algorithm was no larger than when observations from a fixed target in the middle of the void were regularly assimilated.

Noise Handling Capabilities of Multivariate Calibration Methods

The noise handling capabilities of principal component regression (PCR) and partial least squares regression (PLSR) are somewhat disputed issues, especially regarding regressor noise. In an attempt to indicate an answer to the question, this article presents results from Monte Carlo simulations assuming a multivariate mixing problem with spectroscopic data. Comparisons with the best linear unbiased estimator (BLUE) based on Kalman filtering theory are included. The simulations indicate that both PCR and PLSR perform comparatively well even at a considerable regressor noise level. The results are also discussed in relation to estimation of pure spectra for the mixing constituents, i.e. to identification of the data generating system. In this respect solutions to well-posed least squares problems serve as references.

Applications of Kalman filtering to real-time trace gas concentration measurements.

A Kalman filtering technique is applied to the simultaneous detection of NH3 and CO2 with a diode-laser-based sensor operating at 1.53 micrometers. This technique is developed for improving the sensitivity and precision of trace gas concentration levels based on direct overtone laser absorption spectroscopy in the presence of various sensor noise sources. Filter performance is demonstrated to be adaptive to real-time noise and data statistics. Additionally, filter operation is successfully performed with dynamic ranges differing by three orders of magnitude. Details of Kalman filter theory applied to the acquired spectroscopic data are discussed. The effectiveness of this technique is evaluated by performing NH3 and CO2 concentration measurements and utilizing it to monitor varying ammonia and carbon dioxide levels in a bioreactor for water reprocessing, located at the NASA-Johnson Space Center. Results indicate a sensitivity enhancement of six times, in terms of improved minimum detectable absorption by the gas sensor.

A data assimilation method used with an ocean circulation model and its application to the tropical Atlantic

A data assimilation method based on the Kalman-filter theory is presented and tested with the NOAA/GFDL Modular Ocean Model (MOM_2). The method is used to assimilate observed tropical Atlantic Ocean surface and subsurface temperatures obtained from the PIRATA moored buoy array. Comparison between model results with and without assimilation shows that the method works properly, and that it has some advantages in relation to a standard Kalman-filter technique. It provides better agreement with the observed temperature intraseasonal variability, and it does not require linearity of a numerical model.

拡張カルマンフィルタを用いたせん断型制震系の地震応答指定型層剛性設計

A stiffness design method is developed for a shear building with prescribed viscous and/or viscoelastic dampers such that every peak interstory drift would coincide with the specified value. The method consists of the calculation of the peak interstory drift and its sensitivity, and the revalidation of the design variables, respectively based on the Complete Quadratic Combination (CQC) method and the extended Kalman filter theory. Twenty-story shear building models are designed by the developed method on the conditions of different weight and damper distributions. The validity and accuracy of the proposed method are demonstrated by the time-history response analyses on the designed models subjected to a design-spectrum-compatible ground motions. The results indicate that this method is useful in case the damping factor of the structural system is less than 20%.

Loop Gain Identification for Adaptive and Reconfigurable Control

The identification of a linear discrete-time control system’s open-loop gain is addressed. The classical Kalman filter theory for state estimation in linear control systems is extended, and the control system’s loop gain and states are jointly estimated. A rigourous analysis of the measurement situation on hand yields explicit formulae for the loop gain’s unbiased estimate and estimation error’s covariance.

Application of the fokker-planck equation to data assimilation into hydrodynamical models

A numerical ocean model with an original data assimilation technique is considered. The data-assimilation approach is based on the well-known Kalman filter theory, but the method of computation of the covariation matrix is new. This method uses a diffusion stochastic representation of the error (difference between the model and observative values), and then the Fokker-Planck equation is solved for determination of the joint distribution of errors in each of two different space points. Compared to the ordinary Kalman-filter technique, which requires n2 operations, where n is the number of grid points, this method requires only m2 operations, where m is the number of observations. Also, the method does not requiere linearity of the model. This method is considered in conjunction with a thermodynamic ocean model based on the primitive equations. Some model experiments have been carried out. The stability of this technique is examined, and possible applications to other models is also discussed.