Standard Kalman Filter Research Articles

On adaptive filtering for high dimensional systems under parameter uncertainty and its application to satellite data assimilation in oceanography

In this paper, the adaptive filtering theory, recently proposed and developed the authors of present work [1-9] for stochastic, encountered in the field of data as simulation in meteorology and oceanography, is reviewed. Several important questions on numerical estimation og the gain matrix, model reduction, structural choices for the gain, filter stability… are discussed. We show the connections of present approach with a standard Kalman filtering. Adaptive filter is implemented along with a Kalman filtering. Adaptive filter is implemented along with a Kalman filter and standard Newton relation method on the four-layer adiabatic Miami Isopycnical Co-ordinate Ocean Model (MICOM) to produce the estimate for the deep oceanic circulation using assimilate synthetic observations of surface height. Numerical results justify high efficiency of the adaptive filter whose performance is slightly better than that of a Kalman filter due to impossibility to correctly specify the error statistics in a Kalman filter.

A new robust filtering for a GPS/SINS loosely coupled integration system

Integrated global positioning system (GPS)/strap-down inertial navigation system (SINS) systems are widely used for positioning and attitude determination applications. Kalman filtering is the most common method to combine both sensors. A standard Kalman filter (SKF) relies on the correct definition of the measurement and process models and cannot handle measurements with outliers. The H∞ filter can achieve better robustness performance based on minimising the worst-case estimation error. Different to other adaptive KFs, the H∞ filter controls its robustness by a restriction parameter γ. The γ parameter is usually set and fixed by experience. In this paper, an adaptive strategy to gain a time-varying γ is proposed. The construction of the time-varying parameter is based on a two-segment function of an approximate ratio of the traces of the calculated predicted residual's covariance and the theoretical covariance. The test results indicate that the proposed filtering can achieve the expected robustness performance and efficiency more comprehensively when compared with other filtering methods.

Anomaly detection in Industrial Autonomous Decentralized System based on time series

With the ever growing demand of location-independent access to Autonomous Decentralized Systems (ADS), anomaly detection scheme for industrial Ethernet, which highly is satisfied with demanding real-time and reliable industrial applications, becomes one of the most pressing subjects in ADS. In this paper, we present an innovative approach to build a traffic model based on structural time series model for a chemical industry system. A basic structural model that decomposes time series into four items is established by the stationary analysis of industrial traffic. Parameters in the model are identified by the state space model, which is conducted from the training sequence using standard Kalman filter recursions and the EM algorithm. Furthermore, the performance of state space model is evaluated by the experimental results that confirm significant improvement in detection accuracy and the validity of abnormal data localization.

Experimental investigation on target detection and tracking in passive radar using long‐term evolution signal

This study examines the feasibility of the recently deployed wireless communication system, namely, long-term evolution (LTE), for passive radar applications. Specifically, the capability of the system to detect and track a ground moving target is analysed. A theoretical analysis is conducted on the actual LTE signal transmitted in the atmosphere, followed by field experimentation using a moving vehicle with different bistatic geometries. The experiment examines the capability of an LTE-based passive radar to detect a moving vehicle and then track it using a standard Kalman filter. Results show that the LTE-based passive radar can detect and track the ground vehicle at different locations, speeds and trajectories, and localise it in the XY-axis coordinate.

Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

AbstractIn this paper, the stability of the adaptive fading extended Kalman filter with the matrix forgetting factor when applied to the state estimation problem with noise terms in the non–linear discrete–time stochastic systems has been analysed. The analysis is conducted in a similar manner to the standard extended Kalman filter’s stability analysis based on stochastic framework. The theoretical results show that under certain conditions on the initial estimation error and the noise terms, the estimation error remains bounded and the state estimation is stable.The importance of the theoretical results and the contribution to estimation performance of the adaptation method are demonstrated interactively with the standard extended Kalman filter in the simulation part.

An Opportunistic Sensor Scheduling Solution to Remote State Estimation Over Multiple Channels

We consider a sensor scheduling problem where the sensors have multiple choices of communication channel to send their local measurements to a remote state estimator for state estimation. Specifically, the sensors can transmit high-precision data packets over an expensive channel or low-precision data packets, which are quantized in several bits, over some cheap channels. The expensive channel, though being able to deliver more accurate data which leads to good estimation quality at the remote estimator, can only be used scarcely due to its high cost (e.g., high energy consumption). On the other hand, the cheap channel, though having a small cost, delivers less accurate data which inevitably deteriorates the remote estimation quality. In this work we propose a new framework in which the sensors switch between the two channels to achieve a better tradeoff among the communication cost, the estimation performance and the computational complexity, where the two-channel case can be easily extended to a multiple-channel case. We propose an opportunistic sensor schedule which reduces the communication cost by randomly switching among the expensive and cheap channels, and in the meantime maintains low computational complexity while introducing data quantization into the estimation problem. We present a minimum mean square error (MMSE) estimator in a closed-form under the proposed opportunistic sensor schedule. We also formulate an optimization problem to search the best opportunistic schedule with a linear quantizer. Furthermore, we show that the MMSE estimator in the limiting case becomes the standard Kalman filter.

The Tobit Kalman Filter: An Estimator for Censored Measurements

Tobit model censored data arise in multiple engineering applications through saturating sensors, limit-of-detection effects, and image frame effects. In this brief, we introduce a novel formulation of the Kalman filter for Tobit Type 1 censored measurements. Our proposed formulation, called the Tobit Kalman filter, is identical to the standard Kalman filter in the no-censoring case. At or near the censored region, the Tobit Kalman filter utilizes a local approximation of the probability of censoring in order to provide a fully recursive estimate of the state and state error covariance. The additional computational burden of the method compared with the standard Kalman filter is limited to the calculation of $m$ normal probability density functions and $m$ normal cumulative density functions per update, where $m$ is the number of measurements.

Real-time State Estimation for an Adaptive Vibration Energy Harvesting System

Abstract: Moving horizon estimation (MHE) has proven to be an efficient optimization-based technique to address the problem of joint state and parameter estimation with nonlinearities in system dynamics and constraints. This work aims to propose the application of MHE for a specific nonlinear vibration setup, often used to analyze energy harvesting concepts in the laboratory environment. Using a simulated energy harvesting scenario exhibiting a strongly nonlinear behavior, we focus on the estimation of unmeasured system states and a time-varying structural parameter that can be utilized to maximize the harvested energy, or in structural health monitoring. To exploit the recent algorithmic developments in embedded optimization, we implement the proposed scheme on a low-cost embedded computing platform using automatic code generation. A hardware-in-the-loop setup is used in order to investigate the performance of the real-time MHE scheme, and to quantify the associated computational effort. The results demonstrated in this paper, suggest the real-time feasibility of MHE, and certain practical advantages over a standard Kalman filtering in a vibration energy harvesting system.

Underwater Inertial Navigation With Long Baseline Transceivers: A Near-Real-Time Approach

Inertial navigation systems for underwater vehicles are often aided by acoustic time-of-flight positioning schemes. One widely implemented long baseline (LBL) approach uses a ping-response protocol resulting in asynchronous measurements that depend on the state of the vehicle at two time instants. Such aiding measurements that depend on the state at multiple time instants do not fit the model format of the standard extended Kalman filter (EKF) framework. This paper proposes a near-real-time (NRT) Bayesian smoothing framework for the LBL-aided inertial navigation system application. Within this NRT framework, the time interval between LBL cycles is divided into two intervals defined by the ping-response cycles of LBL. The time instant when all the LBL responses are received will be referred to as the NRT point . Between the LBL ping and the NRT point, a traditional real-time EKF is implemented using the inertial measurement unit and all aiding measurements. After the NRT point, when all the LBL responses have been received, an optimal Bayesian trajectory estimator executes. This maximum a posteriori (MAP) estimation includes all the measurement information already collected up to the NRT point of the current LBL cycle. The MAP output is a smoothed trajectory estimate for the period of time between the LBL ping and the NRT point. At the conclusion of this smoothing process, the current EKF estimate is corrected to the optimal MAP solution and propagated to the current time, and then the EKF estimation process continues to the NRT point of the next LBL cycle. At all times, the approach maintains a real-time state estimate suitable for use by the control system. This paper presents the theoretical solution, discusses the implementation, and presents implementation results to illustrate the accuracy and reliability of this NRT approach.

Extended robust Kalman filter for attitude estimation

In this study, the authors deal with inertial measurement units subject to uncertainties. They propose an extended robust Kalman filter (ERKF) in a predictor–corrector form to estimate a rigid body attitude. The filter is developed based on regularisation and penalisation whose approaches present the advantage of encompassing in a unified framework all state and output uncertain parameters of the system. The ERKF is tuned based on two degree of freedom which belong to a certain interval known a -priori, useful for online applications. The attitude estimation system proposed takes into account a rigid body model formulated in terms of quaternions. Experimental results are presented based on a comparative study among the ERKF, the standard extended Kalman filter and an ℋ∞ filter.

Deterministic Mean-Field Ensemble Kalman Filtering

The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $\kappa$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2\kappa$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory.

A Novel Robust Interval Kalman Filter Algorithm for GPS/INS Integrated Navigation

Kalman filter is widely applied in data fusion of dynamic systems under the assumption that the system and measurement noises are Gaussian distributed. In literature, the interval Kalman filter was proposed aiming at controlling the influences of the system model uncertainties. The robust Kalman filter has also been proposed to control the effects of outliers. In this paper, a new interval Kalman filter algorithm is proposed by integrating the robust estimation and the interval Kalman filter in which the system noise and the observation noise terms are considered simultaneously. The noise data reduction and the robust estimation methods are both introduced into the proposed interval Kalman filter algorithm. The new algorithm is equal to the standard Kalman filter in terms of computation, but superior for managing with outliers. The advantage of the proposed algorithm is demonstrated experimentally using the integrated navigation of Global Positioning System (GPS) and the Inertial Navigation System (INS).

Vehicle Tracking in Video using Fractional Feedback Kalman Filter

Object tracking is a challenging and important area of research. The object tracking system must be capable of tracking abrupt variations in object state. Kalman filter is fundamental and widely used as an optimal state estimator in object tracking. With known noise and system parameters, Kalman filter tends to stabilize the gain. However, during sudden transitions of the object tracked, constant gain Kalman filter may diverge. This paper proposes a modified steady-state gain of the Kalman filter, which is achieved by introducing a fractional feedback loop across the Kalman gain. The modified Kalman gain is estimated by minimizing the cost function of the proposed Kalman filter. Results show that the accuracy and robustness of the Kalman filter are significantly improved. The performance of the proposed method is compared with that of the standard Kalman filter, fractional-order Kalman filter, and unscented Kalman filter. In this work, the root mean square error is used as the performance metric. The proposed method has been tested for different datasets, including traffic videos. Experiments show that RMSE is improved upto $17\%$ by using the proposed modified Kalman filter.

An Explicit Data Assimilation Scheme for a Nonlinear Wave Prediction Model Based on a Pseudo-Spectral Method

A robust data assimilation scheme is presented for a wave model to predict evolving nonlinear ocean waves. The Fourier coefficients of the surface elevation and the free surface velocity potential are chosen for state variables and are propagated in time by solving numerically a set of nonlinear evolution equations using a pseudo-spectral method. The numerical solutions are then updated with noise corrupted measurements of the surface elevation with the aid of an explicit Kalman filter for which the time evolution of the error covariance matrix is found explicitly by solving analytically the linearized wave prediction model. After presenting an error analysis for this explicit data assimilation scheme, numerical simulations of the integrated nonlinear wave prediction model for long-crested waves of varying wave steepness are performed by using synthetic data with different noise characteristics. It is shown that the estimated surface wave fields agree well with the true states, and the present data assimilation scheme based on the explicit Kalman filter improves considerably the computational efficiency and stability, in comparison with a standard Kalman filter for which the error covariance matrix is found numerically.

Distributed Kalman filters with adaptive strategy for linear time‐varying interconnected systems

SummaryWe consider the problem of distributed state estimation over a sensor network in which a set of nodes collaboratively estimates the state of a continuous‐time linear time‐varying system. In particular, our work focuses on the benefits of weight adaptation of the interconnection gains in distributed Kalman filters. To this end, an adaptation strategy is proposed with the adaptive laws derived via a Lyapunov‐redesign approach. The justification for the gain adaptation stems from a desire to adapt the pairwise difference of state estimates as a function of their agreement, thereby enforcing an interconnection‐dependent gain. In the proposed scheme, an adaptive gain for each pairwise difference of the interconnection terms is used in order to address edge‐dependent differences in the state estimates. Accounting for node‐specific differences, a special case of the scheme is also presented, where it uses a single adaptive gain in each node estimate and which uniformly penalizes all pairwise differences of state estimates in the interconnection term. The filter gains can be designed either by standard Kalman filter or Luenberger observer to construct the adaptive distributed Kalman filter or adaptive distributed Luenberger observer. Stability of the schemes has been shown, and it is not restricted by the graph topology and therefore the schemes are applicable to both directed and undirected graphs. The proposed algorithms offer a significant reduction in communication costs associated with information flow by the nodes. Finally, numerical studies are presented to illustrate the performance and effectiveness of the proposed adaptive distributed Kalman filters. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Sequential nonlinear tracking filter without requirement of measurement decorrelation

Sequential measurement processing is of benefit to both estimation accuracy and computational efficiency. When the noises are correlated across the measurement components,decorrelation based on covariance matrix factorization is required in the previous methods in order to perform sequential updates properly. A new sequential processing method, which carries out the sequential updates directly using the correlated measurement components, is proposed. And a typical sequential processing example is investigated, where the converted position measurements are used to estimate target states by standard Kalman filtering equations and the converted Doppler measurements are then incorporated into a minimum mean squared error(MMSE)estimator with the updated cross-covariance involved to account for the correlated errors. Numerical simulations demonstrate the superiority of the proposed new sequential processing in terms of better accuracy and consistency than the conventional sequential filter based on measurement decorrelation.

A stochastic unknown input realization and filtering technique

This paper studies the state estimation problem of linear discrete-time systems with unknown inputs which can be treated as a wide-sense stationary process with rational power spectral density, while no other prior information needs to be known. We propose an autoregressive (AR) model based unknown input realization technique which allows us to recover the input statistics from the output data by solving an appropriate least squares problem, then fit an AR model to the recovered input statistics and construct an innovations model of the unknown inputs using the eigensystem realization algorithm. An augmented state system is constructed and the standard Kalman filter is applied for the state estimation. A reduced order model filter is also introduced to reduce the computational cost of the Kalman filter. A numerical example is given to illustrate the procedure.

Vehicle tractive force prediction with robust and windup-stable Kalman filters

Vehicle control systems need to prognosticate future vehicle states in order to improve energy efficiency. This paper compares four approaches that are used to identify the parameters of a longitudinal vehicle dynamics model used for the prediction of vehicle tractive forces. All of the identification approaches build on a standard Kalman filter. Measurement signals are processed using the polynomial function approximation technique to remove noise and compute smooth derivative values of the signals. Experimental results illustrate that the approach using multiple Stenlund–Gustafsson M-Kalman filters (multiple robust and windup-stable Kalman filters) reaches the best performance and robustness in predicting the vehicle tractive forces.

Image noise smoothing using a modified Kalman filter

Noise smoothing is significant in image processing applications. Precipitated by previous research, this paper aims at reducing the noise using the Kalman filter. First, an image model based on Markov random field is built. Then the standard Kalman filter is implemented as a smoother. In order for the innovation to be fully used by the filter, the multi-innovation is introduced to improve the filtering/smoothing performance. Experimental results validate the proposed methods.

A new method to improve estimation of uncertain parameters in the Ensemble Kalman filter by re-parameterization employing prior statistics correction

The Ensemble Kalman Filter (EnKF) is a Monte Carlo based method to assimilate the measurement data sequentially in time. Although, EnKF has some advantages over the other Kalman based methods to deal with non-linear and/or high dimensional reservoir models, it also suffers from deficiency in estimation of non-Gaussian parameters. In this work, we propose a re-parameterization method to handle non-Gaussian parameters via Ensemble Kalman Filter framework. For this purpose, concept of cumulative distribution function transformation has been used. In addition, the statistics of prior information have been aggregated in the state vector in order to capture the prior uncertainties of non-Gaussian parameters. To evaluate the performance of the new method, three estimation examples have been implemented. These examples are designed to evaluate the performance of the proposed method to estimate both local and global parameters. The performance of method in handling non-Gaussian prior as well as capturing the prior uncertainty has been assessed. The results revealed that, the proposed method handled both non-Gaussian prior and capturing of the prior uncertainties of the local and global parameters quite efficiently. Compared to previously conducted researches, new algorithm would considerably reduce the number of required ensemble members to converge to the appropriate solution. Standard ensemble Kalman filter and normal score transformation have been also implemented for each setup; the obtained results show that the new proposed method outperforms the other alternate methods.