Fixed-interval Kalman Smoother Research Articles

Dual-Modal Electrical Imaging of Two-Phase Flow-Experimental Evaluation of the State Estimation Approach.

Accurate measurement of two-phase flow quantities is essential for managing production in many industries. However, the inherent complexity of two-phase flow often makes estimating these quantities difficult, necessitating the development of reliable techniques for quantifying two-phase flow. In this paper, we investigated the feasibility of using state estimation for dynamic image reconstruction in dual-modal tomography of two-phase oil-water flow. We utilized electromagnetic flow tomography (EMFT) to estimate velocity fields and electrical tomography (ET) to determine phase fraction distributions. In state estimation, the contribution of the velocity field to the temporal evolution of the phase fraction distribution was accounted for by approximating the process with a convection-diffusion model. The extended Kalman filter (EKF) and fixed-interval Kalman smoother (FIKS) were used to reconstruct the temporally evolving velocity and phase fraction distributions, which were further used to estimate the volumetric flow rates of the phases. Experimental results on a laboratory setup showed that the FIKS approach outperformed the conventional stationary reconstructions, with the average relative errors of the volumetric flow rates of oil and water being less than 4%. The FIKS approach also provided feasible uncertainty estimates for the velocity, phase fraction, and volumetric flow rate of the phases, enhancing the reliability of the state estimation approach.

Kalman smoother error bounds in the presence of misspecified measurements

The performance of a discrete-time fixed-interval Kalman smoother in the presence of a misspecified measurement equation is considered. An easily calculable numerical bound for the increment in smoothing error energy is provided in an adversarial non-linear setting by formulating a high-dimensional optimization problem which is solvable via its Lagrange dual, a scalar convex optimization problem. The results are validated by an adversarial learner, implemented as a neural network.

Aerosol formation and growth rates from chamber experiments using Kalman smoothing

Abstract. Bayesian state estimation in the form of Kalman smoothing was applied to differential mobility analyser train (DMA-train) measurements of aerosol size distribution dynamics. Four experiments were analysed in order to estimate the aerosol size distribution, formation rate, and size-dependent growth rate, as functions of time. The first analysed case was a synthetic one, generated by a detailed aerosol dynamics model and the other three chamber experiments performed at the CERN CLOUD facility. The estimated formation and growth rates were compared with other methods used earlier for the CLOUD data and with the true values for the computer-generated synthetic experiment. The agreement in the growth rates was very good for all studied cases: estimations with an earlier method fell within the uncertainty limits of the Kalman smoother results. The formation rates also matched well, within roughly a factor of 2.5 in all cases, which can be considered very good considering the fact that they were estimated from data given by two different instruments, the other being the particle size magnifier (PSM), which is known to have large uncertainties close to its detection limit. The presented fixed interval Kalman smoother (FIKS) method has clear advantages compared with earlier methods that have been applied to this kind of data. First, FIKS can reconstruct the size distribution between possible size gaps in the measurement in such a way that it is consistent with aerosol size distribution dynamics theory, and second, the method gives rise to direct and reliable estimation of size distribution and process rate uncertainties if the uncertainties in the kernel functions and numerical models are known.

Electrochemical Model-Based State of Charge and Capacity Estimation for a Composite Electrode Lithium-Ion Battery

Increased demand for hybrid and electric vehicles has motivated research to improve onboard state of charge (SOC) and state of health estimation (SOH). In particular, batteries with composite electrodes have become popular for automotive applications due to their ability to balance energy density, power density, and cost by adjusting the amount of each material within the electrode. SOH algorithms that do not use electrochemical-based models may have more difficulty maintaining an accurate battery model as the cell ages under varying degradation modes, such as lithium consumption at the solid-electrolyte interface or active material dissolution. Furthermore, efforts to validate electrochemical model-based state estimation algorithms with experimental aging data are limited, particularly for composite electrode cells. In this paper, we first present a reduced-order electrochemical model for a composite LiMn <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -LiNi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sub> Mn <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sub> Co <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sub> O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> electrode battery that predicts the surface and bulk lithium concentration of each material in the composite electrode, as well as the current split between each material. The model is then used in dual-nonlinear observers to estimate the cell SOC and loss of cyclable lithium over time. Three different observer types are compared: 1) the extended Kalman filter; 2) fixed interval Kalman smoother; and 3) particle filter. Finally, an experimental aging campaign is used to compare the estimated capacities for five different cells with the measured capacities over time.

An auto-framing method for stochastic process signal by using a hidden Markov model based approach

In this paper, an “auto-framing” method, an algorithmic method to divide stochastic time-series process data into appropriate intervals, is developed based on the approach of hidden Markov model (HMM). While enormous amounts of process time-series data are being measured and collected today, their use is limited by the high costs to gather, store, and analyze them. “Data-framing” refers to the task of dividing stochastic signal data into time frames of distinct patterns so that the data can be stored and analyzed in an efficient manner. Data-framing is typically carried out manually, but doing so can be both laborious and ineffective. For the purpose of automating the data-framing task, stochastic signals of switching patterns are modeled using a hidden Markov model (HMM) based jump linear system (JLS), which switches the stochastic model probabilistically in accordance with the underlying Markov chain. Based on the model, an estimator is constructed to estimate from the collected signal data the state sequence of the underlying Markov chain, which is subsequently used to decide on the framing points. An Expectation Maximization (EM) algorithm, which is composed of two optimal estimators, fixed interval Kalman smoother and Viterbi algorithm, is used to estimate for the state estimation. We demonstrate the effectiveness of the HMM-based approach for auto-framing using simulated data constructed based on real industrial data.

Fixed-Interval Kalman Smoother for Non-Square Descriptor Systems with Correlated Noise

Based on the classical Kalman filtering theory, the state estimation problem is considered for non-square descriptor discrete time stochastic systems. Under Assumptions 1~3, a fixed-Interval Kalman smoother for non-square descriptor systems with correlated noise is given. Some numerical examples illustrate the effectiveness of the proposed algorithm.

Improved elimination of motion artifacts from a photoplethysmographic signal using a Kalman smoother with simultaneous accelerometry

A photoplethysmography (PPG) signal provides very useful information about a subject's hemodynamic status in a hospital or ubiquitous environment. However, PPG is very vulnerable to motion artifacts, which can significantly distort the information belonging to the PPG signal itself. Thus, the reduction of the effects of motion artifacts is an important issue when monitoring the cardiovascular system by PPG. There have been many adaptive techniques to reduce motion artifacts from PPG signals. In the present study, we compared a method based on the fixed-interval Kalman smoother with the usual adaptive filtering algorithms, e.g. the normalized least mean squares, recursive least squares and the conventional Kalman filter. We found that the fixed-interval Kalman smoother reduced motion artifacts from the PPG signal most effectively. Therefore, the use of the fixed-interval Kalman smoother can reduce motion artifacts in PPG, thus providing the most reliable information that can be deduced from the reconstructed PPG signals.

Hidden Gauss-Markov models for signal classification

Continuous-state hidden Markov models (CS-HMMs) are developed as a tool for signal classification. Analogs of the Baum (1972), Viterbi (1962), and Baum-Welch algorithms are formulated for this class of models. The CS-HMM algorithms are then specialized to hidden Gauss-Markov models (HGMMs) with linear Gaussian state-transition and output densities. A new Gaussian refactorization lemma is used to show that the Baum and Viterbi algorithms for HGMMs are implemented by two different formulations of the fixed-interval Kalman smoother. The measurement likelihoods obtained from the forward pass of the HGMM Baum algorithm and from the Kalman-filter innovation sequence are shown to be equal. A direct link between the Baum-Welch training algorithm and an existing expectation-maximization (EM) algorithm for Gaussian models is demonstrated. A new expression for the cross covariance between time-adjacent states in HGMMs is derived from the off-diagonal block of the conditional joint covariance matrix. A parameter invariance structure is noted for the HGMM likelihood function. CS-HMMs and HGMMs are extended to incorporate mixture densities for the a priori density of the initial state. Application of HGMMs to signal classification is demonstrated with a three-class test simulation.

Optimality of variational data assimilation and its relationship with the Kalman filter and smoother

The known properties of equivalence between four-dimensional variational (4D-Var) data assimilation and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. In the linear context, the 4D-Var solution is optimal, not only with respect to the model trajectory segment over the assimilation time interval, but also with respect to any model state at a single observation time level; in the batch processing (cycling 4D-Var) method, the information in 4D-Var is fully transferred from one batch to the next by the background term; 4D-Var allows the processing of observations in subsets, while the final solution is optimal as all observations are processed simultaneously. These properties hold even for models that are imperfect, as well as not invertible. Various properties of equivalence of 4D-Var to the Kalman filter and smoother result from these optimality properties of 4D-Var. Further, we show that the fixed-lag Kalman smoother may also be constructed in an optimal way using a multiple batch-processing 4D-Var approach. While error covariances are crucial for the equivalence, practical techniques for evaluating error covariances in the framework of cycling 4D-Var are discussed.

Adaptive Tuning of Numerical Weather Prediction Models: Simultaneous Estimation of Weighting, Smoothing, and Physical Parameters

In Wahba et al. it was shown how the randomized trace method could be used to adaptively tune numerical weather prediction models via generalized cross validation (GCV) and related methods. In this paper a “toy” four-dimensional data assimilation model is developed (actually one space and one time variable), consisting of an equivalent barotropic vorticity equation on a latitude circle, and used to demonstrate how this technique may be used to simultaneously tune weighting, smoothing, and physical parameters. Analyses both with the model as a strong constraint (corresponding to the usual 4D-Var approach) and as a weak constraint (corresponding theoretically to a fixed-interval Kalman smoother) are carried out. The conclusions are limited to the particular toy problem considered, but it can be seen how more elaborate experiments could be carried out, as well as how the method might be applied in practice. The authors have considered five adjustable parameters, two related to a distributed coefficient in the equivalent barotropic vorticity equation (“physical” parameters), one governing the relative weight given to observations versus forecast, one governing the relative weight given to observations versus goodness of fit to the model (in the weak constraint case), and one governing the damping of high-frequency oscillations in the analysis at the final time point (“smoothing” parameter). The weighting parameters and the smoothing parameter can, if desired, be interpreted as ratios of parameters in prior covariances. Analyses are made with a low-resolution model of the dynamics of the equivalent barotropic vorticity equation given noisy forecast (initial conditions) and noisy wind observations, and compared with nature evolved from exact initial conditions using a high-resolution forward integration. The authors found that these five (carefully chosen) parameters are simultaneously tunable on line, that is, simultaneously with the analysis, and 1) that the analysis is equally and strongly sensitive to both the choice of the observed versus forecast weighting parameter and the choice of the smoothing parameter; 2) that the analysis with the model as a weak constraint, based on the tuned estimate of the parameter governing how close the analysis satisfies the model, is somewhat better than the analysis using the model as a strong constraint, although estimation of this tuning parameter varies much more than the other parameters with replications of the experiment; and 3) good estimates of the physical parameters are obtained; however, these estimates are closer to those that make the model integrated forward with perfect initial conditions best fit nature, and these are not exactly the “true” parameters.

The application of Kalman smoother theory to the estimation of 4DV AR error statistics

Modern atmospheric data assimilation theory is dominated by the four-dimensional variational (4DVAR) and Kalman filter/smoother approaches. Both generate analysis weights (explicitly or implicitly) which are dynamically determined by the assimilation model. A Kalman smoother is basically a generalization of the Kalman filter which can process future observations. In control theory, a generalization of 4DVAR called Pontryagin optimization can account for an imperfect assimilating model. Pontryagin optimization and the fixed-interval Kalman smoother are equivalent when both methods use the same statistical information. We use the equivalence between Pontryagin optimization and the Kalman smoother to examine the effect of the perfect model assumption on the error statistics and analysis weights of the 4DVAR algorithm. This is done by developing the Kalman smoother equations for a very simple assimilating model. A procedure for diagnosing the effect of model error, based on the observational cost function, is also developed. DOI: 10.1034/j.1600-0870.1996.t01-1-00003.x

The independance of forward and backward estimation errors in the two-filter form of the fixed interval Kalman smoother

In this paper it is shown that the Fraser-Potter two-filter form of the fixed interval Kalman smoother can be derived without assuming that the forward and backward estimation errors are independent. In fact, it is shown that when the transformed process noise autocorrelation is nonsingular, the aforementioned independence is a conclusion rather than a hypothesis. Some additional conclusions pertinent to computations are also provided.