Process Noise Variance Research Articles

An Adaptive Strong Tracking Volume Kalman Filter Algorithm

Abstract Based on dental implant technology, this paper studies the position and posture tracking of surgical tools during the treatment of patients. A strong tracking adaptive volume Kalman filter (STF-ACKF) algorithm is proposed to meet the requirements of high accuracy, low time delay and strong robustness due to the complexity of the oral space and the narrow motion environment, to improve the measurement accuracy and reduce the noise interference. The adaptive module is added to the algorithm, and the real-time correction of process noise variance is realized by using MAP estimator [1-3]. The variance of the measured noise [4-5] was estimated using the variational Bayes (VB) method. Finally, the theoretical analysis and simulation results show that the mean error of STF-ACKF on X-axis, Y-axis and yaw angle is reduced by 1.17 mm, 0.76 mm and 0.62 ° respectively, the estimation accuracy and stability are improved significantly.

Ionospheric process noise estimation via single-receiver GNSS data

ABSTRACT To fully exploit the multi-epoch data processing in GNSS parameter estimation, a thorough understanding of the parameters’ temporal behavior is required. This behavior is expressed by a dynamic model whose uncertainty is captured by the parameters’ process noises. The more randomly the temporal behavior deviates from the dynamic model, the larger the process noise variance should be considered. In GNSS positioning, the ionospheric parameters are known to be time-varying, requiring a reliable choice of process noise. In this contribution, we present a data-driven method with which one can estimate ionospheric process noise variances, minimizing the risk of having a suboptimal estimation process. We study the dependency of such process noise variance on solar and geomagnetic activities across various geographic positions. By analyzing several globally distributed datasets, a considerable difference in the ionospheric process noise estimates is observed between daytime and nighttime, emphasizing the importance of modeling ionospheric temporal characteristic in GNSS positioning.

Joint Bayesian estimation of process and measurement noise statistics in nonlinear Kalman filtering

Bayesian filtering aims at sequentially estimating the states and parameters in nonlinear dynamical systems, and finds applications in a variety of engineering fields. Extended and unscented Kalman filter are popular choices for this purpose as they typically achieve a good trade-off between computational complexity and accuracy, while still quantifying posterior uncertainty. However, the accuracy of the identified states and parameters, both in terms of their posterior mean and variance, is heavily influenced by the assumed statistics of both the process and measurement noise terms. Therefore, learning the noise characteristics becomes an essential aspect of Bayesian filtering, but is non-trivial especially in cases where process and measurement noise statistics are jointly non-identifiable. This article proposes a methodology that decomposes measurement and process noise learning in a fully Bayesian setting. For measurement noise estimation, we improve upon a Bayesian method that leverages conjugacy of the inverse-Wishart distribution with respect to the Gaussian likelihood model. For process noise estimation, Bayesian model averaging is coupled with a mutation scheme to compare the performance of competing models while exploring high-probability regions in the space of the unknown process noise variance. The algorithm is first tested on simple numerical models for both state estimation and parameter learning, highlighting the superior accuracy of the proposed measurement noise correction and the benefits of decoupling process and measurement noise learning in non-identifiable settings. Then we demonstrate the usefulness of this algorithm in more challenging problems, namely identification of a multi-degree-of-freedom system under unmeasured excitation, and modeling of the highly nonlinear response of a full-scale bridge pier using experimental data.

Learning capabilities to resolve tilt-translation ambiguity in goldfish.

Spatial orientation refers to the perception of relative location and self-motion in space. The accurate formation of spatial orientation is essential for animals to survive and interact safely with their environment. The formation of spatial orientation involves the integration of sensory inputs from the vestibular, visual, and proprioceptive systems. Vestibular organs function as specialized head motion sensors, providing information regarding angular velocity and linear acceleration via the semicircular canals and otoliths, respectively. However, because forces arising from the linear acceleration (translation) and inclination relative to the gravitational axis (tilt) are equivalent, they are indistinguishable by accelerometers, including otoliths. This is commonly referred to as the tilt - translation ambiguity, which can occasionally lead to the misinterpretation of translation as a tilt. The major theoretical frameworks addressing this issue have proposed that the interpretation of tilt versus translation may be contingent on an animal's previous experiences of motion. However, empirical confirmation of this hypothesis is lacking. In this study, we conducted a behavioral experiment using goldfish to investigate how an animal's motion experience influences its interpretation of tilt vs. translation. We examined a reflexive eye movement called the vestibulo-ocular reflex (VOR), which compensatory-rotates the eyes in response to head motion and is known to reflect an animal's three-dimensional head motion estimate. We demonstrated that the VORs of naïve goldfish do not differentiate between translation and tilt at 0.5 Hz. However, following prolonged visual-translation training, which provided appropriate visual stimulation in conjunction with translational head motion, the VORs were capable of distinguishing between the two types of head motion within 3 h. These results were replicated using the Kalman filter model of spatial orientation, which incorporated the variable variance of process noise corresponding to the accumulated motion experience. Based on these experimental and computational findings, we discuss the neural mechanism underlying the resolution of tilt-translation ambiguity within a context analogous to, yet distinct from, previous cross-axis VOR adaptations.

A reduced-condition-number algorithm for single-frequency precise point positioning based on regularized Kalman filter

Single-frequency receivers are low cost and portable, thus being widely applied in engineering; the extended Kalman filter (EKF) is commonly used to perform single-frequency precise point positioning (SF-PPP). However, the positioning performance of SF-PPP is seriously influenced by various errors. Due to the large process noise and initial variance of the estimated parameters, the weight matrix of state parameters will be ill-conditioned, and since the noise of the pseudo-range is much higher than that of the carrier phase, the weight matrix of observations presents as ill-conditioned. Additionally, the condition number of the normal matrix will jump on the conditions of cycle slip, new emerging satellites, and signal outages. To reduce the condition number of the normal matrix, the regularized Kalman filter (RKF) algorithm is proposed, with additional support for the maximum variance matrix and singular value decomposition, thereby improving the accuracy and stability of SF-PPP. Through static and dynamic experiments, it is found that the proposed method can reduce both the ill-conditioning of the weight matrices of the observations and the state parameters. The condition number of the normal matrix is <500 per epoch, and the convergence time is shortened by >40%. Compared with the SF-PPP using EKF, centimeter-level static positioning accuracies of 1.13, 0.73, and 2.92 cm and decimeter-level kinematic positioning accuracies of 12.5, 10.8, and 27.3 cm in the east, north, and vertical components, respectively, using RKF; this yielded 38.3, 29.8, and 45.2% and 39.6, 41.9, and 21.3% improvement in the static and kinematic scenarios, respectively.

Estimating process noise variance of PPP-RTK corrections: a means for sensing the ionospheric time-variability

Estimating process noise variance of PPP-RTK corrections: a means for sensing the ionospheric time-variability

AN ADAPTIVE KALMAN FILTERING ALGORITHM WITHOUT USING KINEMATIC MODELS

The performance and accuracy of Kalman filter depends on its gain value related to the process noise covariance and the measurement noise variance which may vary according to experimental settings such as noise and sampling time. Thus, setting the appropriate values for the noise variances that fit for a wide range of experimental setting is a challenge for conventional Kalman filter. This paper proposes an adaptive Kalman filter with the adaptive noise variance for velocity estimation without using kinematic model. By applying only the quantized position measurement signal generated from the optical incremental encoder, an adaptive process noise variance is proposed. The experimental results show that the proposed method outperforms the conventional Kalman filter in achieving accurate and smooth velocity estimation without large time delay.

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter.

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

MSE Design of Nearly Constant Velocity Kalman Filters for Tracking Targets With Deterministic Maneuvers

When tracking maneuvering targets with a nearly constant velocity (NCV) Kalman filter with white noise acceleration, the selection of the process noise variance is complicated by the fact that the process noise errors are modeled as white Gaussian in the Kalman filter, while target maneuvers are deterministic or highly correlated in time. In recent years, the deterministic maneuver index was introduced for NCV Kalman filters and used to develop a relationship between the anticipated maximum acceleration of the target and the process noise variance that minimizes the maximum mean squared error (MaxMSE) in position. Lower bounds on the process noise variance that prevent the position MSE from exceeding the measurement noise variance were also expressed in terms of the maximum acceleration and deterministic maneuver index. In this paper, those results are summarized and codified for practical application by the target tracking community. The design methods for NCV Kalman filters with discrete white noise acceleration (DWNA) and continuous white noise acceleration (CWNA) are presented for sustained and brief maneuvers. The application of the design methods to radar tracking of maneuvering targets is also addressed. The effectiveness of the design methods is illustrated via Monte Carlo simulations.

An Improved Particle-Filter-Based Hybrid Optimization Algorithm for IoT Positioning in Uncertain WSNs

With the wide applications of the Internet of Things (IoT) technologies in intelligent manufacturing, production safety, smart cities and other fields, target location awareness has been the primary issue as it can be used to personnel and material positioning, electronic fence settings, daily attendance statistics, and so on. Wireless Sensor Networks (WSNs) positioning can become an indispensable part of these location-based IoT applications that benefit from the ubiquitous sensing and communication ability. For addressing the large positioning errors caused by uncertain WSNs, this paper proposes an improved particle filter-based hybrid optimization (IPFHO) algorithm. After mapping the noisy wireless signal to uncertain target location, the preliminary positioning of mobile target is realized by improved particle filter, whose coarse accuracy over time can be further optimized with use of iterative search method. We evaluate the proposed algorithm under different noise levels, process noise variances and computation times in extensive simulations. The results indicate that the positioning accuracy of proposed algorithm can be effectively improved in the presence of wireless ranging errors and anchor node calibration errors. Compared with the positioning error 0.27m estimated by pure filtering, the positioning error of proposed algorithm can be reduced to 0.19m by combining the filtering and search methods. The platform experimental results, which are consistent with the trend of the simulation results, validate the superior accuracy of proposed algorithm compared with relevant positioning algorithms.

State-of-charge estimation of lithium-ion batteries based on MCC-AEKF in non-Gaussian noise environment

The extended Kalman filter (EKF) has low estimation accuracy for lithium-ion battery SOC under non-Gaussian noise interference. To solve this problem, an adaptive EKF based on maximum correlation-entropy criterion (MCC-AEKF) was proposed. In this algorithm, the maximum correlation entropy criterion is used to obtain accurate estimation results by replacing the overall solution with the local optimal solution. Meanwhile, the adaptive covariance matrix is established to update the process noise variance to reduce the SOC estimation error. MCC-AEKF were used to estimate the SOC based on two kinds of battery data. The experiments show that, under non-Gaussian noise interference and at operating temperature of 25 °C, compared with EKF and MCC-EKF, the estimation accuracy of MCC-AEKF improved by 80.3% and 24.2% separately for battery 1. For battery 2, its estimation accuracy improved by 72.6% and 22.0% separately, and are also the highest at 10 °C and 40 °C. Given wrong initial SOC values, MCC-AEKF can converges to real value within 30s. All results present the good accuracy and robustness of the proposed method under different cases.

Estimation of process noise variances from the measured output sequence with application to the empirical model of type 1 diabetes

This paper presents a novel method for estimating a priori unknown variances of the process noise affecting a general discrete-time stochastic state-space model under the assumption that the elements of the process noise vector are not correlated. To evaluate the proposed method in practice, only the system model and a sufficiently long sequence of output measurements are required, so there is no need for the system state to be measured or the state observer to be involved. In detail, the design of the method is based on the autocorrelation function of the stochastic output component, which is used to form the equivalent linear regression system with respect to the unknown vector of the process noise variances. Since the data-based estimate of the autocorrelation function has to be treated as uncertain, the generalized least squares method is proposed to solve the derived linear regression system, yielding the estimate with minimal variance. To verify the theory, the method is first validated within a case study that considers an artificial linear stochastic system, while this experiment was carried out under controlled statistical conditions. However, the intended target application domain of the method is the empirical model of type 1 diabetes, so another experiment based on virtual diabetic data obtained by simulating a complex physiology-based model that was subjected to input uncertainties was carried out.

An Improved EKF Algorithm for SOC Estimation of Lithium Battery Considering Temperature Effects

In order to improve the estimation accuracy of the state of charge (SOC) of lithium-ion power batteries during use, an estimation method that considers different temperature effects is proposed in this paper. First, the second-order resistor-capacitor (RC) circuit model is selected as the research model based on a comparative analysis of different models. Then, the parameters in the circuit model are identified online in real time under different temperatures to analyze the temperature effects on internal parameters. Lastly, the noise variance can be automatically adjusted according to the environment by using the improved extended Kalman filter (EKF) algorithm with the differential evolution algorithm, which optimizes the fixed process noise variance in the original EKF algorithm. The changing noise variance is substituted in the original EKF algorithm to estimate SOC under cycle working conditions online. The results show that the improved EKF algorithm increases the estimation accuracy of the SOC of lithium batteries. Using the improved EKF algorithm, the estimated value can quickly converge to the actual value at different temperatures, and the SOC error remains within 2% in the whole estimation process.

Robust Dynamic CPU Resource Provisioning in Virtualized Servers

We present robust dynamic resource allocation mechanisms to allocate application resources meeting Service Level Objectives (SLOs) agreed between cloud providers and customers. In fact, two filter-based robust controllers, i.e., <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> filter and Maximum Correntropy Criterion Kalman filter (MCC-KF), are proposed. The controllers are self-adaptive, with process noise variances and covariances calculated using previous measurements within a time window. In the allocation process, a bounded client mean response time ( <inline-formula><tex-math notation="LaTeX">$\mathop {\mathrm{mRT}}$</tex-math></inline-formula> ) is maintained. Both controllers are deployed and evaluated on an experimental testbed hosting the RUBiS (Rice University Bidding System) auction benchmark web site. The proposed controllers offer improved performance under abrupt workload changes, shown via rigorous comparison with current state-of-the-art. On our experimental setup, the Single-Input-Single-Output (SISO) controllers can operate on the same server where the resource allocation is performed; while Multi-Input-Multi-Output (MIMO) controllers are on a separate server where all the data are collected for decision making. SISO controllers take decisions not dependent to other system states (servers), albeit MIMO controllers are characterized by increased communication overhead and potential delays. While SISO controllers offer improved performance over MIMO ones, the latter enable a more informed decision making framework for resource allocation problem of multi-tier applications.

Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter.

This paper evaluates the performance of robust adaptive tracking techniques with the direct-state Kalman filter (DSKF) used in modern digital global navigation satellite system (GNSS) receivers. Under the assumption of a well-known Gaussian distributed model of the states and the measurements, the DSKF adapts its coefficients optimally to achieve the minimum mean square error (MMSE). In time-varying scenarios, the measurements’ distribution changes over time due to noise, signal dynamics, multipath, and non-line-of-sight effects. These kinds of scenarios make difficult the search for a suitable measurement and process noise model, leading to a sub-optimal solution of the DSKF. The loop-bandwidth control algorithm (LBCA) can adapt the DSKF according to the time-varying scenario and improve its performance significantly. This study introduces two methods to adapt the DSKF using the LBCA: The LBCA-based DSKF and the LBCA-based lookup table (LUT)-DSKF. The former method adapts the steady-state process noise variance based on the LBCA’s loop bandwidth update. In contrast, the latter directly relates the loop bandwidth with the steady-state Kalman gains. The presented techniques are compared with the well-known state-of-the-art carrier-to-noise density ratio ()-based DSKF. These adaptive tracking techniques are implemented in an open software interface GNSS hardware receiver. For each implementation, the receiver’s tracking performance and the system performance are evaluated in simulated scenarios with different dynamics and noise cases. Results confirm that the LBCA can be successfully applied to adapt the DSKF. The LBCA-based LUT-DSKF exhibits superior static and dynamic system performance compared to other adaptive tracking techniques using the DSKF while achieving the lowest complexity.

Fault detection in a three-tank hydraulic system using unknown input observer and extended Kalman filter

This paper presents the problem of fault diagnosis in a three-tank hydraulic system. A mathematical model of the system is developed in order to apply two different observing algorithms. Unknown Input Observer (UIO) and Extended Kalman Filter (EKF) have been used to detect and isolate actuator and sensor faults. For Unknown Input Observer (UIO), residuals are calculated from the measured and estimated output according to the eigenvalues of the system after processed by Linear Matrix Inequality (LMI). Extended Kalman filter uses process and measurement noise variances for state estimation. Unknown Input Observer and Extended Kalman Filter's performance in fault estimation and isolation is evaluated under different scenarios. Using Extended Kalman Filter (EKF), faults can be diagnosed effectively in the presence of noise, while Unknown Input Observer (UIO) is working better in the absence of noise, and simulation results illustrate that clearly.

An intelligent particle filter with resampling of multi-population cooperation

The particle filter (PF) has excellent estimation performance for nonlinear non-Gaussian systems. However, this method misleads the results due to sample impoverishment and the sensitivity on the initial values of state and process noise variance. To overcome this, an intelligent PF method with a resampling of multi-population cooperation is presented in this paper. Firstly, an intelligent resampling mechanism based on a circular collaborative structure is proposed. In this mechanism, the particles are divided into multiple populations, which are generated by importance densities with different initial state values and variances individually. Secondly, a collaborative strategy based on Gaussian mutation is designed to improve particle diversity so as to raise estimation accuracy of PF. This strategy reserves the high-weight particles in each population, and replaces the low-weight ones with the particles generated by Gaussian mutation on high-weight particles in the previous population based on the circular collaborative structure. Finally, three systems are introduced to test the performance of the proposed method. The results illustrate that the proposed method can effectively improve the estimation accuracy and robustness of PF when the initial values of state and variance of process noise are unknown compared with three existing methods.

Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity

In multipath-assisted simultaneous localization and mapping (SLAM), the geometric association of specular multipath components based on radio signals with environmental features is used to simultaneously localize user equipment and map the environment. We must contend with two notable model parameter uncertainties in multipath-assisted SLAM: process noise and clutter intensity. Knowledge of these two parameters is critically important to multipath-assisted SLAM, the uncertainty of which will seriously affect the SLAM accuracy. Conventional multipath-assisted SLAM algorithms generally regard these model parameters as fixed and known, which cannot meet the challenges presented in complicated environments. We address this challenge by improving the belief propagation (BP)-based SLAM algorithm and proposing a robust multipath-assisted SLAM algorithm that can accommodate model mismatch in process noise and clutter intensity. Specifically, we describe the evolution of the process noise variance and clutter intensity via Markov chain models and integrate them into the factor graph representing the Bayesian model of the multipath-assisted SLAM. Then, the BP message passing algorithm is leveraged to calculate the marginal posterior distributions of the user equipment, environmental features and unknown model parameters to achieve the goals of simultaneous localization and mapping, as well as adaptively learning the process noise variance and clutter intensity. Finally, the simulation results demonstrate that the proposed approach is robust against the uncertainty of the process noise and clutter intensity and shows excellent performances in challenging indoor environments.

Industry Tip: Picking the Minimum Process Noise Variance for Your NCV Track Filter

When tracking maneuvering targets with a nearly constant velocity (NCV) Kalman filter with discrete white noise acceleration, the selection of the process noise variance is complicated by the fact that the process noise errors are modeled as white Gaussian, while target maneuvers are deterministic or highly correlated in time. In this industry tip, guidance on picking the minimum process noise variance for the NCV Kalman filter is provided.

Dynamic measurement of the airflow rate in a two-zones dwelling, from the CO2 tracer gas-decay method using the Kalman filter

The measurement of natural airflows is challenging, because of low pressures involved in natural ventilation, and because of their variability. The tracer gas decay method is the most used method to assess natural airflows. It is, however, not adapted to multi-zone dwellings. A protocol based on the decay method is tested here involving the Kalman filter, and a correction procedure of concentrations. Both intend to allow the measurement of the global airflow-rate of a multi-zone passive-stack ventilated dwelling. The correction procedure of concentrations inhibits infiltrations that would have an impact on the accuracy of the tracer decay method implemented in a multi-zone dwelling. The Kalman filter allows measuring a dynamic airflow-rate, while the background tracer gas concentration varies which would bias the conventional decay method. It is a key issue when implementing the CO2 decay method in a multi-zone dwelling as, depending on its sources, the CO2 concentration in any room of the dwelling is likely to vary. The robustness of the proposed protocol was tested through a parametrical analysis in laboratory twincells and it showed a significantly lower sensitivity compared to the conventional 2-points decay method. Maximum deviations on the airflow-rate were systematically lower from 4 to 11 points. The concentration correction procedure allowed increasing the accuracy up to 190 points. The ability of the Kalman filter to assess the dynamic airflow-rate was questioned. It was strongly influenced by the process noise variance associated to state parameters of the Kalman filter, which are defined by the experimenter.