Filtering Equations Research Articles

A Dual-State Filter for a Relative Velocity Aiding Strapdown Inertial Navigation System

The strapdown inertial navigation system that uses the aiding velocities from a relative log is usually affected by the ocean current. Therefore, a dual-state filter algorithm to alleviate the effect from unknown ocean current is proposed in this article. Generally speaking, the ocean-current velocity is relatively stable, in other words, two adjacent samples over a short period can be considered to be unchanged. Consequently, our algorithm chooses the difference between the two adjacent sample points from the log as the external measurement information of the integrated navigation system, where the effect from ocean-current velocity will be alleviated in the filter. Nevertheless, the previous state (the previous velocity error) will exist in the filter updating process. The common Kalman filter algorithm cannot cope with this condition, so we rederived filter equations regarding the dual-state filter based on the previous state and the current state. The final results of simulation and experiment demonstrate that the dual-state filter algorithm can alleviate the effect from unknown ocean current.

Distributed state estimation through co‐acting Kalman filters

AbstractDistributing calculations of a central Kalman filter requires subsystem level expressions for the propagation and update steps of the Kalman filter. It is difficult to obtain subsystem level expressions due to the inverse term present in the update step. In this manuscript, a non‐iterative way of decomposing the inverse of a matrix is presented. This decomposition allows rewriting the update equations of the Kalman filter subsystem‐wise. Subsequently, a Co‐acting Kalman Filter (CoKF) is proposed using these decomposed central Kalman filter equations to perform distributed state estimation. The convergence of the CoKF algorithm is established under the assumption that each subsystem is observable. Two variants of the proposed CoKF, namely (m‐CoKF and p‐CoKF), suitable for applications on opposite ends of computation and communication resource spectrum, are presented along with the trade‐offs involved. A comparison of the proposed method with existing distributed Kalman filters is also presented. The proposed CoKF algorithm is implemented on a standard wireless sensor network example with 200 nodes. The simulation results demonstrate the accuracy of the proposed CoKF algorithm relative to the central Kalman filter.

On the convergence and stability of fractional singular Kalman filter and Riccati equation

In our recent paper [1], we have developed a simple algorithm for the state estimation of discrete-time linear stochastic fractional-order singular (FOS) systems based on the deterministic least squares method. In this paper, we shall consider the asymptotic properties of the obtained generalized Riccati equation and the associated state estimator. The approach is based on the analysis in detail of the derived filter and algebraic Riccati equation by generalizing significantly the results for standard causal systems, in which provide us conditions for the stability and convergence of the fractional singular Kalman filter (FSKF). Also, conditions under which the Riccati equation has a unique positive semi-definite (PSD) solution are given. Towards this, stability studies of discrete-time FOS state-space systems are investigated, and detectability and stabilizability criterions have been derived. Some examples are performed and verified using numerical simulations in order to illustrate the given analysis.

Angular velocity fusion of the microelectromechanical system inertial measurement unit array based on extended Kalman filter with correlated system noises

The paper proposes an angular velocity fusion method of the microelectromechanical system inertial measurement unit array based on the extended Kalman filter with correlated system noises. In the proposed method, an adaptive model of the angular velocity is built according to the motion characteristics of the vehicles and it is regarded as the state equation to estimate the angular velocity. The signal model of gyroscopes and accelerometers in the microelectromechanical system inertial measurement unit array is used as the measurement equation to fuse and estimate the angular velocity. Due to the correlation of the state and measurement noises in the presented fusion model, the traditional extended Kalman filter equations are optimized, so as to accurately and reliably estimate the angular velocity. By simulating angular rates in different motion modes, such as constant and change-in-time angular rates, it is verified that the proposed method can reliably estimate angular rates, even when the angular rate has been out of the microelectromechanical system gyroscope measurement range. And results show that, compared with the traditional angular rate fusion method of microelectromechanical system inertial measurement unit array, it can estimate angular rates more accurately. Moreover, in the kinematic vehicle experiments, the performance advantage of the proposed method is also verified and the angular rate estimation accuracy can be increased by about 1.5 times compared to the traditional method.

Impact of total variation minimization in volume rendering visualization of breast tomosynthesis data

Background and objectiveTotal Variation (TV) minimization algorithms have achieved great attention due to the virtue of decreasing noise while preserving edges. The purpose of this work is to implement and evaluate two TV minimization methods in 3D. Their performance is analyzed through 3D visualization of digital breast tomosynthesis (DBT) data with volume rendering. MethodsBoth filters were studied with real phantom and one clinical DBT data. One algorithm was applied sequentially to all slices and the other was applied to the entire volume at once. The suitable Lagrange multiplier used in each filter equation was studied to reach the minimum 3D TV and the maximum contrast-to-noise ratio (CNR). Imaging blur was measured at 0° and 90° using two disks with different diameters (0.5 mm and 5.0 mm) and equal thickness. The quality of unfiltered and filtered data was analyzed with volume rendering at 0° and 90°. ResultsFor phantom data, with the sequential filter, a decrease of 25% in 3D TV value and an increase of 19% and 30% in CNR at 0° and 90°, respectively, were observed. When the filter is applied directly in 3D, TV value was reduced by 35% and an increase of 36% was achieved both for CNR at 0° and 90°. For the smaller disk, variations of 0% in width at half maximum (FWHM) at 0° and a decrease of about 2.5% for FWHM at 90° were observed for both filters. For the larger disk, there was a 2.5% increase in FWHM at 0° for both filters and a decrease of 6.28% and 1.69% in FWHM at 90° with the sequential filter and the 3D filter, respectively. When applied to clinical data, the performance of each filter was consistent with that obtained with the phantom. ConclusionsData analysis confirmed the relevance of these methods in improving quality of DBT images. Additionally, this type of 3D visualization showed that it may play an important complementary role in DBT imaging. It allows to visualize all DBT data at once and to analyze properly filters applied to all the three dimensions.Concise AbstractTotal Variation (TV) minimization algorithms are one compressed sensing technique that has achieved great attention due to the virtue of decrease noise while preserve edges transitions. The purpose of this work is to solve the same TV minimization problem in DBT data, by studying two 3D filters. The obtained results were analyzed at 0° and 90° with a 3D visualization through volume rendering. The filters differ in their application. One considers a slice-by-slice optimization, sequentially traversing all slices of the data. The other considers the intensity values of adjacent slices to make this optimization on each voxel. The performance of each filter was also tested with a clinical case. The results obtained were very encouraging with a significantly increased contrast to noise ratio at 0° and 90° and a small reduction in blur at 90° (slight reduction of the out-of-plane artifact).

A multi‐tone central divided difference frequency tracker with adaptive process noise covariance tuning

SummaryThe problem of real‐time frequency estimation of nonstationary multi‐harmonic signals is important in many applications. In this paper, we propose a novel multi‐frequency tracker based on a state‐space representation of the signal with Cartesian filters and the second‐order central divided difference filter (CDDF), which improves the performance of the extended Kalman filter (EKF) by using Stirling's interpolation method to approximate the mean and covariance of the state vector. A crucial element of the method is the adaptive scaling of the process noise covariance matrix appearing in the filter equations, as a function of the innovation sequence, which tunes the accuracy‐reactivity trade‐off of the filter. The proposed solution is evaluated against two approaches from the literature, namely the factorized adaptive notch filter (FANF) and the extended Kalman filter frequency tracker (EKFFT). Several experiments emphasize the estimation accuracy of the proposed method as well as the improved robustness with respect to initial errors and input signal complexity. The presented method appears to be particularly efficient with rapidly varying frequencies, thanks to the update mechanism that adjusts the filter parameters based on the amplitude of the estimation error.

A Minimum Energy Filter for Distributed Multirobot Localisation

We present a new approach to the cooperative localisation problem by applying the theory of minimum energy filtering. We consider the problem of estimating the pose of a group of mobile robots in an environment where robots can perceive fixed landmarks and neighbouring robots as well as share information with others over a communication channel. Whereas the vast majority of the existing literature applies some variant of a Kalman Filter, we derive a set of filter equations for the global state estimate based on the principle of minimum energy filtering. We show how the filter equations can be decoupled and the calculations distributed among the robots in the network without requiring a central processing node. Finally, we provide a demonstration of the filter’s performance in simulation.

Multi-Frequency Data Fusion for Attitude Estimation Based on Multi-Layer Perception and Cubature Kalman Filter

This paper proposes multi-frequency inertial and visual data fusion for attitude estimation. The proposed strategy is based on the locally weighted linear regression (LWLR), multi-layer perception (MLP), and cubature Kalman filter (CKF). First, we analyze the discrepant-frequency and the attitude divergence problems. Second, we construct the filter equation for the visual and inertial data and attitude differential equation for inertial-only data, which are used to estimate the attitude in time series. Third, we employ LWLR to compute the vision discrepancies between actual vision data and fitted vision data. The vision discrepancy is used as the input of MLP training. In MLP, the discrepancy is used as weights of the sums through the activation function of the hidden layer. To address the divergence problem, which is inherent in a multi-frequency fusion, the MLP is utilized to compensate for the inertial-only data. Finally, experimental results on different environments of pseudo-physical simulations show the superior performance of the proposed method in terms of the accuracy of attitude estimation and divergence capability.

A Kriging-based Interacting Particle Kalman Filter for the simultaneous estimation of temperature and emissivity in Infra-Red imaging

Temperature estimation through infrared thermography is facing the lack of knowledge of the observed material’s emissivity. The derivation of the physical equations lead to an ill-posed problem. A new Kriged Interacting Particle Kalman Filter is proposed. A state space model relates the measurements to the temperature and the Kalman filter equations yield a filter tracking the temperature over time. Moreover, a particle filter associated to Kriging prediction is interacting with a bank of Kalman filters to estimate the time-varying parameters of the system. The efficiency of the algorithm is tested on a simulated sequence of infrared thermal images.

Stochastic Wiener filter in the white noise space

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

Numerical Gaussian process Kalman filtering

In this manuscript we introduce numerical Gaussian process Kalman filtering (GPKF). Numerical Gaussian processes have recently been developed to simulate spatiotemporal models. The contribution of this paper is to embed numerical Gaussian processes into the recursive Kalman filter equations. This embedding enables us to do Kalman filtering on infinite-dimensional systems using Gaussian processes. This is possible because i) we are obtaining a linear model from numerical Gaussian processes, and ii) the states of this model are by definition Gaussian distributed random variables. Convenient properties of the numerical GPKF are that no spatial discretization of the model is necessary, and manual setting up of the Kalman filter, that is fine-tuning the process and measurement noise levels by hand is not required, as they are learned online from the data stream. We showcase the capability of the numerical GPKF in a simulation study of the advection equation.

Derivation of Modified Input Estimation Filter Using Bayesian Framework

A new approach to derive the modified input estimation is proposed using Bayesian framework. Tracking manoeuvring targets is a tough problem and many algorithms are proposed to deal with it. Among these algorithms, modified input estimation (MIE) technique has been proved to be an effective method. MIE technique treats acceleration as additive input term in the state vector and estimates the original state and acceleration vectors simultaneously with a standard Kalman filter. Analogous to standard Kalman filter, however, the MIE filter equations which have a modification on Kalman gain is given without a cohesive derivation. Using a Bayesian framework, we will present a conceptually cohesive roadmap that starts at first principles and leads directly to derivation of MIE filter equations.

Design of a Quantum Projection Filter

This article develops a quantum projection filtering approach to approximating the quantum filter equation based on It $\hat{\rm o}$ stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.

Primordial power spectrum and cosmology from black-box galaxy surveys

ABSTRACT We propose a new, likelihood-free approach to inferring the primordial matter power spectrum and cosmological parameters from arbitrarily complex forward models of galaxy surveys where all relevant statistics can be determined from numerical simulations, i.e. black boxes. Our approach, which we call simulator expansion for likelihood-free inference (selfi), builds upon approximate Bayesian computation using a novel effective likelihood, and upon the linearization of black-box models around an expansion point. Consequently, we obtain simple ‘filter equations’ for an effective posterior of the primordial power spectrum, and a straightforward scheme for cosmological parameter inference. We demonstrate that the workload is computationally tractable, fixed a priori, and perfectly parallel. As a proof of concept, we apply our framework to a realistic synthetic galaxy survey, with a data model accounting for physical structure formation and incomplete and noisy galaxy observations. In doing so, we show that the use of non-linear numerical models allows the galaxy power spectrum to be safely fitted up to at least kmax = 0.5 h Mpc−1, outperforming state-of-the-art backward-modelling techniques by a factor of ∼5 in the number of modes used. The result is an unbiased inference of the primordial matter power spectrum across the entire range of scales considered, including a high-fidelity reconstruction of baryon acoustic oscillations. It translates into an unbiased and robust inference of cosmological parameters. Our results pave the path towards easy applications of likelihood-free simulation-based inference in cosmology. We have made our code pyselfi and our data products publicly available at http://pyselfi.florent-leclercq.eu.

Wingman-Based Estimation and Guidance for a Sensorless PN-Guided Pursuer

A novel wingman-based estimation and guidance concept is proposed for a sensorless pursuer. The pursuer is guided towards a maneuvering aerial target using proportional navigation (PN) guidance law. The wingman is assumed to acquire bearings-only measurements of the target and to accurately track the wingman-pursuer relative position. The pursuer-target relative states, needed for the pursuer guidance law implementation, are estimated from the available data to the wingman. The proposed state estimator is implemented using extended Kalman filter equations and transformed wingman's measurements into the moving pursuer frame. Analytical observability analysis of the proposed wingman-based measuring concept suggests an optimal wingman trajectory in terms of the wingman-pursuer relative geometry. The resulting wingman trajectory ensures maximum observability of the pursuer-target line-of-sight (LOS) angle, which is a crucial parameter needed for the PN guidance law implementation. The resulting trajectory can be directly related to the well-known LOS guidance concept. Monte Carlo simulation results validate the analytical findings and demonstrate the potential of the proposed concept.

Kalman Filter for Noise Reducer on Sensor Readings

Most systems nowadays require high-sensitivity sensors to increase its system performances. However, high-sensitivity sensors, i.e. accelerometer and gyro, are very vulnerable to noise when reading data from environment. Noise on data-readings can be fatal since the real measured-data contribute to the performance of a controller, or the augmented system in general. The paper will discuss about designing the required equation and the parameter of modified Standard Kalman Filter for filtering or reducing the noise, disturbance and extremely varying of sensor data. The Kalman Filter equation will be theoretically analyzed and designed based on its component of equation. Also, some values of measurement and variance constants will be simulated in MATLAB and then the filtered result will be analyzed to obtain the best suitable parameter value. Then, the design will be implemented in real-time on Arduino to reduce the noise of IMU (Inertial Measurements Unit) sensor reading. Based on the simulation and real-time implementation result, the proposed Kalman filter equation is able to filter signal with noises especially if there is any extreme variation of data without any information available of noise frequency that may happen to sensor- reading. The recommended ratio of constants in Kalman Filter is 100 with measurement constant should be greater than process variance constant.

Using Kalman filter algorithm for short-term traffic flow prediction in a connected vehicle environment

We develop a Kalman filter for predicting traffic flow at urban arterials based on data obtained from connected vehicles. The proposed algorithm is computationally efficient and offers a real-time prediction since it invokes the connected vehicle data just before the prediction period. Moreover, it can predict the traffic flow for various penetration rates of connected vehicles (the ratio of the number of connected vehicles to the total number of vehicles). At first, the Kalman filter equations are calibrated using data derived from Vissim traffic simulator for different penetration rates, different fluctuating arrival rates of vehicles and various signal settings. Then the filter is evaluated for a variety of traffic scenarios generated in Vissim simulator. We evaluate the performance of the algorithm for different penetration rates under several traffic situations using some statistical measures. Although many of the previous prediction methods depend highly on data from fixed sensors (i.e., loop detectors and video cameras), which are associated with huge installation and maintenance costs, this study provides a low-cost mean for short-term flow prediction only based on the connected vehicle data.

A Cortisol-Based Energy Decoder for Investigation of Fatigue in Hypercortisolism.

Hormones play a fundamental role in homeostasis. We develop a state-space model relating the body's internal energy to cortisol hormone secretions. Cortisol is secreted in pulses and follows a 24 h circadian rhythm. Secretory event timings carry important information regarding internal feedback signaling taking place, as do the upper and lower serum cortisol levels. We relate an internal energy state variable to cortisol pulse timings and to the upper and lower serum cortisol envelopes. We derive Bayesian filter equations for state estimation and use the Expectation-Maximization algorithm for model parameter recovery. Results on multi-day simulated data show circadian energy variations in healthy subjects and non-circadian fluctuations throughout 24 h periods in patient models suffering from hypercortisolism. The results shed new light on why patients diagnosed with excess cortisol disorders frequently experience symptoms of daytime fatigue and sleep disturbances at night. The state-space model is also an important first step towards the design of closed-loop controllers for treating hormone-related disorders in a manner that closely emulates the body's own pulsatile feedback mechanisms.

An Enhanced Fusion Strategy for Reliable Attitude Measurement Utilizing Vision and Inertial Sensors

In this paper, we present a radial basis function (RBF) and cubature Kalman filter (CKF) based enhanced fusion strategy for vision and inertial integrated attitude measurement for sampling frequency discrepancy and divergence. First, the multi-frequency problem of the integrated system and the reason for attitude divergence are analyzed. Second, the filter equation and attitude differential equation are constructed to calculate attitudes separately in time series when visual and inertial data are available or when there are only inertial data. Third, attitude errors between inertial and vision are sent to the input layer of RBF for training. After this, through the activation function of the hidden layer, the errors are transferred to the output layer for weighting the sums, and the training model is established. To overcome the problem of divergence inherent in a multi-frequency system, the well-trained RBF, which can output the attitude errors, is utilized to compensate the attitudes calculated by pure inertial data. Finally, semi-physical simulation experiments under different scenarios are performed to validate the effectiveness and superiority of the proposed scheme in accurate attitude measurements and enhanced anti-divergence capability.

Optimal design of LCL filter in grid‐connected inverters

As an essential part in technologies for energy storage systems (ESSs) or renewable energy systems (RESs), grid-connected inverters need power passive filters to meet grid regulations. As typical passive filters, L filter and LCL filter are employed. Although LCL filter is more cost-effective than L filter, a design of LCL filter is more complicated than L filter. LCL filter has three filter elements: inverter-side inductor, grid-side inductor, and filter capacitor. To design the three elements for LCL filter, three or more simultaneous equations should be required, which means that three or more design target should exist. Since conventional design methods for LCL filter do not suggest clear targets and do follow try & error method which cannot be matched exactly with design targets, the conventional methods cannot be optimal design. In this study, a filter inductance ratio to minimise total filter inductance, a filter admittance to meet grid regulation and characteristic impedance for low current stress of switch stack are suggested as design goals. Optimal design equation is proposed to meet the three design goals. The proposed method can solve unique filter elements for LCL filter without iterative try & error. The design method is evaluated through simulation and experiment.