Classical Kalman Filter Research Articles

State estimation with multi-level vector quantisation and communication uncertainty

In this paper, we design a state estimation algorithm for vector state-vector measurement systems over wireless sensor networks subject to bandwidth limitation and communication uncertainty. With the aid of Mahalanobis transformation, vector measurement innovations are decorrelated into the normalised ones to facilitate parallel quantisation. Then, taking account of Gaussian channel noises, a generalised multi-level quantisation mechanism and the minimum mean square error (MMSE) estimator are jointly designed, where optimal quantisation parameters can be solved by minimising the estimation error covariance with given quantisation level. The proposed MMSE estimator not only has a similar recursive structure as the classical Kalman filter, but also dramatically reduces the sensor-to-estimator communication requirement with only a slight deterioration of estimation performance. The combined effect of quantisation mechanism and communication uncertainty on estimation performance is also discussed. Finally, Monte Carlo simulation results illustrate the effectiveness and efficiency of the proposed quantised estimator.

Identification of joint structural state and earthquake input based on a generalized Kalman filter with unknown input

Accurate and real-time information of structural states and earthquake input is the prerequisite for structural seismic safety assessment and vibration control. When the earthquake input to a structure is not measured, it has been tried to identify structural state and the unknown earthquake input using the measured structural responses as an inverse problem. However, only structural absolute acceleration responses instead of the relative ones can be measured in this case, which limits the real-time performance of the existing Kalman filter with unknown input (KF-UI). In this paper, a generalized Kalman filter with unknown input (GKF-UI) is proposed to identify structural states and unknown earthquake inputs in real-time. Structural motion equations are established in the relative coordinate system and the observation equations are structural absolute acceleration-only measurements. The analytical derivation of the proposed GKF-UI is a direct extension of the classical Kalman filter (KF). It is also proven that the identification by the proposed GKF-UI using the absolute acceleration-only measurements under unknown earthquake excitation does not lead to the so-called drifted results in the previous identifications under unknown external excitations. Moreover, structural modeling errors can be considered in the identification by the proposed method. Then, it is extended to investigate the identification of the structural state of high-rise shear buildings and unknown earthquake inputs by combining the proposed GKF-UI with the modal expansion. Hybrid dynamic motion equations in modal space and observation equations in physical space are established to both reduce the state dimension and keep the one-dimension of unknown earthquake input instead of the increased number of unknown modal inputs. In addition, absolute accelerations are used in the observation equations, which avoids the dilemma that absolute accelerations cannot be converted into relative modal accelerations in the relative coordinate system under unknown earthquake input. Some numerical examples are used to verify the proposed method. Moreover, a shaking table test is adopted to further validate the performances of the proposed method.

Modeling time-varying brain networks with a self-tuning optimized Kalman filter.

Brain networks are complex dynamical systems in which directed interactions between different areas evolve at the sub-second scale of sensory, cognitive and motor processes. Due to the highly non-stationary nature of neural signals and their unknown noise components, however, modeling dynamic brain networks has remained one of the major challenges in contemporary neuroscience. Here, we present a new algorithm based on an innovative formulation of the Kalman filter that is optimized for tracking rapidly evolving patterns of directed functional connectivity under unknown noise conditions. The Self-Tuning Optimized Kalman filter (STOK) is a novel adaptive filter that embeds a self-tuning memory decay and a recursive regularization to guarantee high network tracking accuracy, temporal precision and robustness to noise. To validate the proposed algorithm, we performed an extensive comparison against the classical Kalman filter, in both realistic surrogate networks and real electroencephalography (EEG) data. In both simulations and real data, we show that the STOK filter estimates time-frequency patterns of directed connectivity with significantly superior performance. The advantages of the STOK filter were even clearer in real EEG data, where the algorithm recovered latent structures of dynamic connectivity from epicranial EEG recordings in rats and human visual evoked potentials, in excellent agreement with known physiology. These results establish the STOK filter as a powerful tool for modeling dynamic network structures in biological systems, with the potential to yield new insights into the rapid evolution of network states from which brain functions emerge.

SVD-based factored-form Cubature Kalman Filtering for continuous-time stochastic systems with discrete measurements

In this paper, a singular value decomposition (SVD) approach is developed for implementing the cubature Kalman filter. The discussed estimator is one of the most popular and widely used method for solving nonlinear Bayesian filtering problem in practice. To improve its numerical stability (with respect to roundoff errors) and practical reliability of computations, the SVD-based methodology recently proposed for the classical Kalman filter is generalized on the nonlinear filtering problem. More precisely, we suggest the SVD-based solution for the continuous–discrete cubature Kalman filter and design two estimators: (i) the filter based on the traditionally used Euler–Maruyama discretization scheme; (ii) the estimator based on advanced Itô-Taylor expansion for discretizing the underlying stochastic differential equations. Both estimators are formulated in terms of SVD factors of the filter error covariance matrix and belong to the class of stable factored-form (square-root) algorithms. The new methods are tested on a radar tracking problem.

Adjustable Scaling Parameters for State of Charge Estimation for Lithium-Ion Batteries Using Iterative Multiple UKFs

In this paper, one unscented Kalman filter with adjustable scaling parameters is proposed to estimate the state of charge (SOC) for lithium-ion batteries, as SOC is most important in monitoring the latter battery management system. After the equivalent circuit model is applied to describe the lithium-ion battery charging and discharging properties, a state space equation is constructed to regard SOC as its first state variable. Based on this state space model about SOC, one state estimation problem corresponding to the nonlinear system is established. In implementing the unscented Kalman filter, state estimation is influenced by the scaling parameter. Then, one criterion function is constructed to choose the scaling parameter adaptively by minimizing this criterion function. To extend one single unscented Kalman filter with adjustable scaling parameters to multiple module estimation, one improved unscented Kalman filter is advised based on iterative multiple models. Generally, the main contributions of this paper consist in two folds: one is to introduce a selection strategy for the scaling parameter adaptively, and the other is to combine iterative multiple models and a single unscented Kalman filter with adjustable scaling parameters. Finally, two simulation examples confirm that our unscented Kalman filter with adjustable scaling parameters and its improved iterative form are better than the classical Kalman filter; i.e., our obtained SOC estimation error converges to zero.

Generalized algorithms for the identification of seismic ground excitations to building structures based on generalized Kalman filtering under unknown input

The exact information of seismic excitation and structural state is a prerequisite for structural seismic safety assessment and vibration control. When the seismic excitation to a structure is not measured, the seismic excitation can be identified as an inversed problem from measured structural responses. Although some relevant approaches have been developed, there are certain limitations or drawbacks in the existing approaches. To circumvent these problems, two generalized algorithms are proposed for the identification of seismic ground excitation to multi-story and tall buildings, respectively. When the seismic ground excitation to a structure is not measured, the data measured by a structural health monitoring system are structural absolute responses. So the structural motion equation in the absolute coordinate system is derived, in which the unknown seismic ground excitation is treated as unknown external force acting on the structure. First, the identification of unknown seismic excitations to multi-story building structures is studied. A generalized Kalman filtering under unknown input is proposed for the identification of structural state and unknown seismic excitation without the observation of structural absolute acceleration responses at the location of unknown external force. The derivation of the proposed generalized Kalman filtering under unknown input is based on the classical Kalman filter, but is more general than the existing identification approaches based on Kalman filter with unknown input in the deployments of accelerometers in the building structure. Then, it is extended to explore the identification of unknown seismic excitations to tall building structures. To avoid substructural identification from the top to bottom in a sequential manner, the motion equation in absolute coordinate system is reduced by modal expansion. Moreover, instead of the identification of unknown modal forces in previous approaches, the seismic excitation is directly identified without increasing the number of unknown forces. To demonstrate the proposed algorithms, numerical examples of identifying seismic excitations to a 6-story shear building and an 18-story tall building are investigated.

Distributed fusion estimation for multisensor systems with non-Gaussian but heavy-tailed noises

Student’s t distribution is a useful tool that can model heavy-tailed noises appearing in many practical systems. Although t distribution based filter has been derived, the information filter form is not presented and the data fusion algorithms for dynamic systems disturbed by heavy-tailed noises are rarely concerned. In this paper, based on multivariate t distribution and variational Bayesian estimation, the information filter, the centralized batch fusion, the distributed fusion, and the suboptimal distributed fusion algorithms are derived, respectively. The centralized fusion is given in two forms, namely, from t distribution based filter and the proposed t distribution based information filter, respectively. The distributed fusion is deduced by the use of the newly derived information filter, and it has been demonstrated to be equivalent to the centralized batch fusion. The suboptimal distributed fusion is obtained by a parameter approximation from the derived distributed fusion to decrease the computation complexity. The presented algorithms are shown to be the generalization of the classical Kalman filter based traditional algorithms. Theoretical analysis and exhaustive experimental analysis by a target tracking example show that the proposed algorithms are feasible and effective.

An Improved Pedestrian Ttracking Method Based on Wi-Fi Fingerprinting and Pedestrian Dead Reckoning

Wi-Fi based positioning has great potential for use in indoor environments because Wi-Fi signals are near-ubiquitous in many indoor environments. With a Reference Fingerprint Map (RFM), fingerprint matching can be adopted for positioning. Much assisting information can be adopted for increasing the accuracy of Wi-Fi based positioning. One of the most adopted pieces of assisting information is the Pedestrian Dead Reckoning (PDR) information derived from inertial measurements. This is widely adopted because the inertial measurements can be acquired through a Commercial Off The Shelf (COTS) smartphone. To integrate the information of Wi-Fi fingerprinting and PDR information, many methods have adopted filters, such as Kalman filters and particle filters. A new methodology for integration of Wi-Fi fingerprinting and PDR is proposed using graph optimization in this paper. For the Wi-Fi based fingerprinting part, our method adopts the state-of-art hierarchical structure and the Penalized Logarithmic Gaussian Distance (PLGD) metric. In the integration part, a simple extended Kalman filter (EKF) is first used for integration of Wi-Fi fingerprinting and PDR results. Then, the tracking results are adopted as initial values for the optimization block, where Wi-Fi fingerprinting and PDR results are adopted to form an concentrated cost function (CCF). The CCF can be minimized with the aim of finding the optimal poses of the user with better tracking results. With both real-scenario experiments and simulations, we show that the proposed method performs better than classical Kalman filter based and particle filter based methods with both less average and maximum positioning error. Additionally, the proposed method is more robust to outliers in both Wi-Fi based and PDR based results, which is commonly seen in practical situations.

C/N0 Estimator Based on the Adaptive Strong Tracking Kalman Filter for GNSS Vector Receivers.

The carrier-to-noise ratio (C/N0) is an important indicator of the signal quality of global navigation satellite system receivers. In a vector receiver, estimating C/N0 using a signal amplitude Kalman filter is a typical method. However, the classical Kalman filter (CKF) has a significant estimation delay if the signal power levels change suddenly. In a weak signal environment, it is difficult to estimate the measurement noise for CKF correctly. This article proposes the use of the adaptive strong tracking Kalman filter (ASTKF) to estimate C/N0. The estimator was evaluated via simulation experiments and a static field test. The results demonstrate that the ASTKF C/N0 estimator can track abrupt variations in C/N0 and the method can estimate the weak signal C/N0 correctly. When C/N0 jumps, the ASTKF estimation method shows a significant advantage over the adaptive Kalman filter (AKF) method in terms of the time delay. Compared with the popular C/N0 algorithms, the narrow-to-wideband power ratio (NWPR) method, and the variance summing method (VSM), the ASTKF C/N0 estimator can adopt a shorter averaging time, which reduces the hysteresis of the estimation results.

An improved Kalman filter time scale algorithm for atomic clock noise variation

Kalman filter time scale algorithm is a method of real-time estimating atomic clock state. It is of great practical value in the time-keeping work. Reliable Kalman filter time scale algorithm requires a reliable atomic clock state model, a random model and a reasonable estimation method. However, it is difficult to construct accurate state model when the noises of atomic clock change. The random model is generally based on the prior statistical information about atomic clock noises, and the prior statistical information may be distorted. In the process of time scale calculation, the noises of atomic clocks need estimating in the Kalman filter time scale algorithm, which is quantified according to the intensity of the noise. With the change of the external environment or aging of atomic clock, the noise intensity may change, resulting in the disturbance of atomic clock state estimation in the Kalman filter time scale algorithm, which further affects the accuracy and stability of the time scale. On the other hand, the error of the noise intensity estimation of atomic clocks will also affect the performance of time scale. Therefore, it is necessary to control the disturbance caused by the variation of noise intensity or the estimation error of noise intensity. In this regard, an adaptive factor is introduced to improve the Kalman filter time scale algorithm, and another adaptive factor is introduced into the state prediction covariance matrix in Kalman filter time scale algorithm. And the values of the two adaptive factors are calculated in real time by using statistics to control the growth of the state prediction covariance. The disturbance of state estimation of atomic clock is reduced, and the accuracy and stability of time scale are improved. In this paper, the sampling interval of simulated data and the measured data are 300 s and 3600 s respectively. The simulated data and measured data are used to calculate the overlapping Allan deviations of the time scale. The results show that the improved Kalman filter time scale algorithm can improve the stability of the sampling time more than 14400 s compared with classical Kalman filter time scale algorithm, and affect the stability of the sampling time less than 14400 s. The degree of influence is related to the weight algorithm of atomic clock. The measured data in this paper are treated by the “predictability” weighting algorithm, which guarantees the long-term stability of time scale. So the simulated data and measured data show that compared with classical Kalman filter time scale algorithm, the improved Kalman filter clock time scale algorithm can improve the accuracy and the long-term stability of time scale.

SVD-based state and parameter estimation approach for generalized Kalman filtering with application to GARCH-in-Mean estimation

In this paper, a gradient-based adaptive filtering technique for unknown state and parameter estimation is proposed for some extensions of classical state-space models: (i) linear time-invariant multiple-input, multiple-output (LTI MIMO) systems, and (ii) linear pairwise Markov models (PMMs) with the related pairwise Kalman filter (PKF). The solution to robust adaptive filtering problem is grounded in the singular value decomposition (SVD) recently adopted for the classical Kalman filter. In contrast to our previous result, the new methods are designed for the case of thin matrices (pre-arrays) involved in the filtering recursions and sensitivities calculation. Additionally, the paper discusses a time-varying variance models’ calibration by the proposed innovative adaptive filters. In particular, the state-space approach for estimating GARCH-in-Mean(1,1) is explored.

An improved Kalman filter for joint estimation of structural states and unknown loadings

The classical Kalman filter (KF) provides a practical and efficient way for state estimation. It is, however, not applicable when the external excitations applied to the structures are unknown. Moreover, it is known the classical KF is only suitable for linear systems and can\'t handle the nonlinear cases. The aim of this paper is to extend the classical KF approach to circumvent the aforementioned limitations for the joint estimation of structural states and the unknown inputs. On the basis of the scheme of the classical KF, analytical recursive solution of an improved KF approach is derived and presented. A revised form of observation equation is obtained basing on a projection matrix. The structural states and the unknown inputs are then simultaneously estimated with limited measurements in linear or nonlinear systems. The efficiency and accuracy of the proposed approach is verified via a five-story shear building, a simply supported beam, and three sorts of nonlinear hysteretic structures. The shaking table tests of a five-story building structure are also employed for the validation of the robustness of the proposed approach. Numerical and experimental results show that the proposed approach can not only satisfactorily estimate structural states, but also identify unknown loadings with acceptable accuracy for both linear and nonlinear systems.

Chandrasekhar-Based Maximum Correntropy Kalman Filtering With the Adaptive Kernel Size Selection

This technical note is aimed to derive the Chandrasekhar-type recursion for the maximum correntropy criterion (MCC) Kalman filtering (KF). For the classical KF, the first Chandrasekhar difference equation was proposed at the beginning of 1970s. This is the alternative to the traditionally used Riccati recursion and it yields the so-called fast implementations known as the Morf-Sidhu-Kailath-Sayed KF algorithms. They are proved to be computationally cheap because of propagating the matrices of a smaller size than n × n error covariance matrix in the Riccati recursion. The problem of deriving the Chandrasekhar-type recursion within the MCC estimation methodology has never been raised yet in engineering literature. In this technical note, we do the first step and derive the Chandrasekhar MCC-KF estimators for the case of adaptive kernel size selection strategy, which implies a constant scalar adjusting weight. Numerical examples substantiate a practical feasibility of the newly suggested MCC-KF implementations and correctness of the presented theoretical derivations.

Hyperbolic SVD‐based Kalman filtering for Chandrasekhar recursion

The problem of numerical instability of the classical Kalman filter (KF) still remains one of the most important topics in engineering literature. For improving its robustness with respect to roundoff errors, the singular value decomposition (SVD) methodology has been proposed for implementing the underlying classical KF Riccati recursion. In this study, SVD-based filtering is derived for an alternative KF mechanisation that is based on the so-called Chandrasekhar recursion and yields a family of fast KF implementations. The new methodology involves hyperbolic SVD (HSVD) factorisation rather than usual SVD utilised in the Riccati-based filtering. The results of numerical study indicate that the HSVD-based filtering strategy outperforms the conventional Chandrasekhar-based KF while solving ill-conditioned state estimation problem. Together with the existed Cholesky-based algorithms, they are the preferred implementations when solving applications with high reliability requirements within the class of fast Chandrasekhar-based KF implementations.

Factored-form Kalman-like implementations under maximum correntropy criterion

Abstract The maximum correntropy criterion (MCC) methodology is recognized to be a robust filtering strategy with respect to outliers and shown to outperform the classical Kalman filter (KF) for estimation accuracy in the presence of non-Gaussian noise. However, the numerical stability of the newly proposed MCC-KF estimators in finite precision arithmetic is seldom addressed. In this paper, a family of factored-form (square-root) algorithms is derived for the MCC-KF and its improved variant, respectively. The family traditionally consists of three factored-form implementations: (i) Cholesky factorization-based algorithms, (ii) modified Cholesky, i.e. UD-based methods, and (iii) the recently established SVD-based filtering. All these strategies are commonly recognized to enhance the numerical robustness of conventional filtering with respect to roundoff errors and, hence, they are the preferred implementations when solving applications with high reliability requirements. Previously, only Cholesky-based IMCC-KF algorithms have been designed. This paper enriches a factored-form family by introducing the UD- and SVD-based methods as well. A special attention is paid to array algorithms that are proved to be the most numerically stable and, additionally, suitable for parallel implementations. The theoretical properties are discussed and numerical comparison is presented for determining the most reliable implementations.

A simulation smoother for long memory time series with correlated and heteroskedastic additive noise

The purpose of this paper is to extend the work of So (1999) by accommodating heteroskedasticity in long memory processes and correlation in disturbances. We propose an alternative representation of long memory processes from which we develop filtering equations, prediction densities and a simulation smoother for latent state variables as in the classical Kalman filter. We illustrate the simulation smoother introduced in this paper by estimating a class of long memory models with heteroskedastic correlated additive noises.

A New Adaptive High-Order Unscented Kalman Filter for Improving the Accuracy and Robustness of Target Tracking

In target tracking, the tracking process needs to constantly update the data information. For maneuvering target, model mismatch and loss of high-order moment information disrupt the accuracy of the state estimation. In this paper, an adaptive high-order unscented Kalman filter (AHUKF) algorithm is proposed for the case of errors occurring in the capturing of model dynamic behavior using the classical unscented Kalman filter (UKF) algorithm. By introducing the free parameter, the analytical solution of the high-order unscented transformation (UT) was obtained, the basis for choosing free parameters was analyzed, and the stability of the algorithm was discussed. A method for obtaining the optimal adaptive factor based on the prediction residual estimation covariance matrix was proposed, which reduces the influence of the dynamic model error and was applied to the target tracking model. In this paper, the proposed AHUKF is applied to a target tracking problem with state mutation, different sampling intervals, and different turn rates, respectively. Simulation results for target tracking illustrate that the proposed algorithm is more accurate and robust than the UKF, high-order unscented Kalman filter (HUKF) and adaptive unscented Kalman filter (AUKF).

Using the local ensemble Transform Kalman Filter for upper atmospheric modelling

The Advanced Ensemble electron density (Ne) Assimilation System (AENeAS) is a new data assimilation model of the ionosphere/thermosphere. The background model is provided by the Thermosphere Ionosphere Electrodynamics General Circulation Model (TIE-GCM) and the assimilation uses the local ensemble transform Kalman filter (LETKF). An outline derivation of the LETKF is provided and the equations are presented in a form analogous to the classic Kalman filter. An enhancement to the efficient LETKF implementation to reduce computational cost is also described. In a 3 day test in June 2017, AENeAS exhibits a total electron content (TEC) RMS error of 2.1 TECU compared with 5.5 TECU for NeQuick and 6.8 for TIE-GCM (with an NeQuick topside).

Economic-mathematical model for predicting financial market dynamics

Study purpose.Existing approaches to forecasting dynamics of financial markets, as a rule, reduce to econometric calculations or technical analysis techniques, which in turn is a consequence of preferences among specialists, engaged in theoretical research and professional market participants, respectively. The main study purpose is developing a predictive economic-mathematical model that allows combining both approaches. In other words, this model should be estimated using traditional methods of econometrics and, at the same time, take into account the impact on the pricing process of the effect of clustering participants on behavioral patterns, as the basis of technical analysis. In addition, it is necessary that the created economic-mathematical model should take into account the phenomenon of existing historical trading levels and control the influence they exert on price dynamics, when it falls into local areas of these levels. Such analysis of price behavior patterns in certain areas of historical repeating levels is a popular approach among professional market participants. Besides, an important criterion of developing model’s potential applicability by a wide range of the interested specialists is its general functional form’s simplicity and, in particular, its components.Materials and methods. In the study, the market of the pound sterling exchange rate against the US dollar (GBP/USD) for the whole period of 2017 was chosen as the considered financial series, in order to forecast it. The presented economic-mathematical model was estimated by classical Kalman filter with an embedded neural network. The choice of these assessment tools can be explained by their wide capabilities in dealing with non-stationary, noisy financial market time series. In addition, applying Kalman filter is a popular technique for estimation local-level models, which principle was implemented in the newly model, proposed in article.Results. Using chosen approach of simultaneous applying Kalman filter and artificial neural network, there were obtained statistically significant estimations of all model’s coefficients. The subsequent model application on GBP/USD series from the test dataset allowed demonstrating its high predictive ability comparing with added random walk model, in particular judging by percentage of correct forecast directions. All received results have confirmed that constructed model allows effectively taking into account structural features of considered market and building good forecasts of future price dynamics.Conclusion. The study was focused on developing and improving apparatus of forecasting financial market prices dynamics. In turn, economic-mathematical model presented in that paper can be used both by specialists, carrying out theoretical studies of pricing process in financial markets, and by professional market participants, forecasting the direction of future price movements. High percentage of correct forecast directions makes it possible to use proposed model independently or as a confirmatory tool.

Examining the Kalman filter in the field of noise and interference with the non-Gaussian distribution

We have developed a sequential recursive Kalman Filter algorithm to filter data in the field of the non-Gaussian noise distribution to be used in measurement instruments. A special feature of the constructed Kalman Filter algorithm to filter data with the non-Gaussian noises is the absence of a need to determine a priori the statistical characteristics of noise. The applicability of the developed Kalman filtering procedure was tested by processing different distribution laws: the Cauchy, Pareto noises, normal and logistic distributions. The effectiveness of the devised filtering procedure is confirmed by applying the filter when processing experimental data with different laws of noise distribution. We have conducted approbation of the developed procedure for the Kalman filtering based on data obtained experimentally, with respect to the superposition of noise distribution laws. The a priori estimate for a filtering error when the number of iterations exceeds 30 tends to zero. The devised filtering procedure employing the Kalman filter could be used when performing the metrological certification of measuring instruments under industrial conditions. Under such circumstances, measuring information could become noisy due to various noises, including those that are not governed by the Gaussian distribution law. The filter could be used when processing data from control systems over state parameters, implemented on the principle of a magnitude threshold control. The applied aspect of the scientific result obtained implies the possibility of extending the scope of application of the classic Kalman filter in measurement instruments. This is a prerequisite for the development of a generic filtering algorithm using the Kalman filter.