Nonlinear Filtering Research Articles

Systematic use of nonlinear data filtering methods in forecasting tasks

The article describes an approach to the systematic use of nonlinear data filtering methods in tasks of intelligent data analysis and machine learning. The concepts of filtering and non-linear filtering are considered. The analysis of modern methods of optimal and probabilistic nonlinear filtering of statistical data and the peculiarities of their application in solving the problems of estimating the states of dynamic systems is carried out. The application of the Kalman filter and its variants for solving nonlinear filtering problems is analysed. The classification of nonlinear filtering methods is given. In the basis of the classification are digital, optimal and probabilistic filters. Non-recursive and recursive digital filters are studied. The formulation of the problem of optimal filtering based on the Kalman filter is considered. The filtering equation for a free dynamic system based on the state space model of a discrete system is given. The extended Kalman filter and its modifications are considered. The Bayesian method of estimating the state of a nonlinear stochastic system is presented. The problem of linear and nonlinear probabilistic filtering is considered. Three filters are considered as examples of probabilistic filters: an unscented Kalman filter, a point mass filter, and a granular filter. The granular filtering algorithm and its modifications are considered in detail. The architecture of the information-analytical system for solving forecasting problems has been developed. The system consists of the following main components: user interface, information storage subsystem, data analysis and pre-processing subsystem, modelling subsystem, forecast construction and evaluation subsystem, visualization subsystem. As an example of forecasting based on the systematic use of non-linear filtering methods, the task of forecasting the prices of Google shares is considered. A comparison of the quality assessment results of basic models and forecast values without filtering and with different options for applying filters was carried out. To improve the quality of forecasting based on prepared data and based on nonlinear filtering methods, a method based on combined forecasts was used to solve the forecasting problem. The results of forecasting using the combined model are presented

Representing uncertainty in limited-area data assimilating ocean models

A limited-area ocean prediction system is developed to acquire forecast error covariances related to uncertainty in atmospheric forcing and turbulent mixing using perturbed model parameters within an Ensemble Kalman Filter (EnKF). The system performs sequential data assimilation delivering realistic ocean state estimation and forecasts. It is initialised to observations using the EnKF, the hybrid-EnKF and Ensemble Optimal Interpolation (EnOI). It has higher resolution than the parent global model, includes tides and assimilates altimetric sea level anomaly to constrain the offshore mesoscale circulation. Dynamic ensemble spread introduced by parameter uncertainty shows agreement with error estimates obtained from forecast innovation statistics. Hybrid EnKF offers improvements to both EnKF and EnOI, having smaller analysis increments and forecast errors. The ensemble mean of the hybrid-EnKF is more accurate than any member due to the introduction of parameterised error which behaves as additive inflation in the hybrid EnKF system. This indicates the parameterisation acts as a non-linear filter for model forecast error growth. To demonstrate consistency, experiments are carried out for various regions that differ in their oceanographic situations. The hybrid EnKF data assimilation system enables practical use of dynamic ensembles that capture the flow dependent errors which are mixed with static or climatological covariances for situations where model error is systematically under-estimated.

Stochastic filtering under model ambiguity

In this paper, we study a non-linear filtering problem in the presence of signal model uncertainty. The model ambiguity is characterized by a class of probability measures from which the true one is taken. After interchanging the order of extremum problems by using the min-max theorem, we find that the uncertain filtering problem can be converted to a weighted conditional mean-field optimal control problem. Further, we characterize the ambiguity filter and prove its unique existence.

Road slope estimation for heavy-duty vehicles under the influence of multiple source factors in real complex road environments

Accurate road slope information is of paramount importance for optimizing vehicle control and enhancing energy efficiency. In the context of heavy-duty vehicles (HDVs), the precise estimation of road slope presents a formidable challenge. This challenge is rooted in the coupling between vehicle mass and road slope, the robustness of road slope estimation methods across diverse driving styles, and the generalizability of algorithms to real-world road scenarios. To address these challenges, this paper proposes an innovative road slope estimation scheme that comprehensively considers the influences of multiple source factors in a real-world HDV transportation scenario. Firstly, in order to solve the problem of road slope estimation error under complex road types and different driving styles, a complex road slope estimation method based on the nonlinear predictive filter (NPF) combined with the Cubature Kalman filter (CKF) is constructed. To eliminate the problem of vehicle power fluctuation in special traffic scenarios, we propose a special scenario road slope estimation method based on the combination of a low-pass filter (LPF) and the NPF_CKF algorithm. To address the error problem when using low-cost Inertial Measurement Unit (IMU) sensors, we designed a Kalman filter (KF)-based multi-sensor fusion correction method. Finally, we conduct real-world experiments on complex roads at the test site to verify the accuracy of the proposed estimation method and its insensitivity to model parameters. The robustness and applicability of the proposed method are also verified in free-driving experiments conducted in real-world road scenarios.

Strongly robust computational ghost imaging based on nearest neighbor filtering

Computational ghost imaging is a new optical imaging technique that obtains a reconstructed image of a target from a set of illumination patterns, but the quality of the reconstructed image is affected by ambient light noise. In this paper, two nearest neighbor filtered illumination patterns generated based on nonlinear noise filtering algorithm are proposed, which can effectively reduce the effect of ambient light in ghost imaging. When additional noise is added, compared with the reconstructed images of random illumination pattern, Hadamard illumination pattern and linear filter illumination pattern, the SNR of the reconstructed images of two different neighbor filtering illumination patterns is increased by 470 %, 1863 %, 75 % and 517 %, 2022 %, 90 %, respectively. In simulated noise environments, neighbor filtering illumination patterns reduce most of the effects of noise and results in high quality reconstructed images for a variety of target objects. In the experiments, these new illumination patterns has a strong robustness, and the reconstructed images obtained are sharper and contain more detailed information than random illumination pattern, Hadamard illumination pattern, and linearly filtered illumination pattern. Nearest neighbor filtering illumination patterns apply the nonlinear filtering method to computational ghost imaging and obtains excellent imaging results, which provides a new denoising idea for computational ghost imaging.

Adaptive fixed-time containment control of uncertain underactuated underwater vehicles under dynamic event-driven mechanism

This paper presents a dynamic event-driven fixed-time design strategy for the three-dimensional containment control of underactuated underwater vehicles under a directed topology. Our main contribution is the development of a novel error transformation approach based on adaptive multi-leader estimation to address the fixed-time containment (FTC) control problem in the presence of underactuation input and unknown leaders’ velocity. An adaptive FTC control law using stabilizing auxiliary variables and nonlinear filters is constructed based on event-driven conditions. The dynamic thresholds, which depend on containment errors, are presented to efficiently reduce communication resources required for implementing the FTC controller. According to Lyapunov theory, it is shown that the trajectories of the followers converge to the convex hull spanned by multiple leader signals within a fixed time while ensuring the avoidance of the Zeno phenomenon. Finally, the effectiveness of the suggested control approach is verified through simulation results.

Statistical estimation and nonlinear filtering in environmental pollution

Statistical estimation and nonlinear filtering in environmental pollution

Loudspeaker cabinet design by topology optimization

Using material distribution-based topology optimization, we optimize the bandpass design of a loudspeaker cabinet targeting low frequencies. The objective is to maximize the loudspeaker’s output power for a single frequency as well as a range of frequencies. To model the loudspeaker’s performance, we combine a linear electromechanical transducer model with a computationally efficient hybrid 2D–3D model for sound propagation. The adjoint variable approach computes the gradients of the objective function with respect to the design variables, and the Method of Moving Asymptotes (MMA) solves the topology optimization problem. To manage intermediate values of the material indicator function, a quadratic penalty is added to the objective function, and a non-linear filter is used to obtain a mesh independent design. By carefully selecting the target frequency range, we can guide the optimization algorithm to successfully generate a loudspeaker design with the required bandpass character. To the best of our knowledge, this study constitutes the first successful attempt to design the interior structure of a loudspeaker cabinet using topology optimization.

Study of non-linear filter-based algorithms for active noise control of machinary noises

Sometimes, machinary noise comprises of non-linear deterministic noise. Therefore, it is expected that for such cases of noises, non-linear filter based active noise control algorithms can provide improved performance in terms of reduction of noise. Various non-linear filter based algorithms explored in this paper are: Volterra filtered-x least mean square algorithm, Hammerstein algorithm, Fourier algorithm utilising Fourier non-linear filter, even mirror Fourier non-linear filter and CN filter, bilinear algorithm, Legendre algorithm, filtered-s least mean square using functional link artifical neural network (FLANN) filter, generalised FLANN, exponential FLANN and recursive FLANN. Noise from various workshop machines like band saw, compressor and welding process were recorded using a microphone and utilised to study the algorithms. Two cases of primary paths, (a) linear primary path and (b) non-linear primary path with cubic non-linearity, are considered for simulation study. Performance of various algorithms is evaluated based on steady-state residual noise.

Rational b-spline adaptive filter for active noise control of machinary noises

Spline adaptive filter has a different architecture than linear-in-parameter nonlinear filters like Volterra filter. There are various kinds of splines like Bezier spline, B-spline and rational B-spline. Rational B spline has weights associated with control points and hence, provides more flexibility than B-spline to create a curve. This paper presents development of rational b-spline adaptive filter for active noise control. Updating equations to update control points and weights of control points are derived. A simulation study is presented to study performance of proposed rational B-spline adaptive filter to reduce machinary noises. Performance of proposed rational B-spline filter are compared with B-spline filter and Bezier filter for active noise control of experimentally recorded noises of band-saw, CNC machine and compressor

Fixed-time nonlinear-filter-based consensus control for nonlinear multiagent systems

This paper investigates the fixed-time consensus control problem for a class of nonlinear multiagent systems (MASs) based on dynamic event-triggered mechanism. In contrast to the already-existing fixed-time control schemes, which cannot address the singularity problem, by constructing a virtual signal and using inequality technology, a new processing method is provided to solve such a problem. In addition, a novel fixed-time nonlinear filter that converges in a fixed time is introduced, which effectively solves the “explosion of complexity” problem existed in backstepping. Moreover, to improve the utilization of resources, an event-triggered mechanism with internal dynamic variables is proposed. It is worth noting that some existing static event-triggered mechanisms are special cases of the dynamic ones. On this basis, an effective fixed-time event-triggered control scheme is developed for MASs, which minimizes the data transmission as much as possible while ensuring the system performance. More importantly, the scheme can ensure the convergence of the system in a fixed time. Finally, simulation results are shown to support the validity of the developed approach.

Large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations

In this paper, we study large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations. First of all, we establish the large deviation principle for the space-distribution dependent Zakai equation by a weak convergence approach. Then based on the obtained result and the relationship between the space-distribution dependent Zakai equation and the space-distribution dependent Kushner-Stratonovich equation, the large deviation principle for the latter is proved.

Robust Bayesian state and parameter estimation framework for stochastic dynamical systems with combined time-varying and time-invariant parameters

We consider the joint state and parameter estimation for dynamical systems having both time-varying and time-invariant parameters. For systems with time-invariant parameters, it has previously been demonstrated that although nonlinear filters may be used for concurrent state and parameter estimation, employing the Markov Chain Monte Carlo (MCMC) algorithm for estimating time-invariant parameters with nested nonlinear filters for state estimation provides more reliable estimates. Following established methods for parameter estimation using filters, we augment the state vector to include the original system states in addition to the subset of the parameters that are known to vary in time. Conventionally, both time-varying and time-invariant parameters would be appended in the state vector, and thus for the purpose of estimation, both would free to vary in time. However, allowing parameters that are known to be time-invariant to change in time introduces non-physical dynamics into the original system. To avoid inducing artificial dynamics, the time-invariant parameters are estimated using MCMC. Furthermore, by estimating the time-invariant parameters by MCMC, the dimensionality of the augmented state is smaller and the nonlinearity in the ensuing state space model will tend to be weaker compared to the conventional approach. The time-varying parameters are perturbed by random noise processes, and the amplitude of the artificial noise is estimated alongside the time-invariant system parameters using MCMC. Estimating the noise strength in this procedure circumvents the need for manual tuning. We illustrate the above-described approach for a simple dynamical system in which some model parameters are time-varying, while the remaining parameters are time-invariant. We adopt the extended Kalman filter (EKF) for state estimation and the estimation of the time-varying system parameters, but reserve the task of estimating time-invariant parameters, including the model noise strength and the artificial noise strength associated with the time-varying parameter to the MCMC algorithm.

Estimation of DSGE Models with the Effective Lower Bound

We propose a new approach for the efficient and robust Bayesian estimation of medium- and large-scale DSGE models with occasionally binding constraints. At its core lies the Ensemble Kalman filter, a novel nonlinear recursive filter, which allows for fast likelihood approximations even for models with large state spaces. We combine the filter with a computationally efficient solution method for piece-wise linear models a state-of-the-art MCMC sampler. Using artificial data, we demonstrate that our approach accurately captures the true parameters of models with a lower bound on nominal interest rates, even with very long lower bound episodes. We use the approach to analyze the US business cycle dynamics until the Covid-19 pandemic, with a focus on the long lower bound episode after the Global Financial Crisis.

A regularized lattice Boltzmann model with filter for multiphase flow with diffusion-dominated mass transfer considering two-film theory

Complex flow, considering the interfacial mass transfer with the two-film theory, is always encountered in critical industrial processes. The phase-field lattice Boltzmann method (PFLBM) coupling with the revised Fick's law mass transfer convection–diffusion equation (CDE) is a practical approach to predict the bulk concentration distribution in two-phase flows. However, solutions of concentration have oscillations and even diverge near the sharp gradient when the relaxation time of governing equations is close to 0.5 (i.e., diffusion-dominated). In this paper, an integrated PFLBM model considering two-phase flow and interfacial mass transfer with a new filtering algorithm and collision operator was built to extend the wider range of the existing model for the two-film CDE with an extremely low diffusion coefficient. First, the two-film mass transfer model from our team was furthermore developed with the second-order formation to meet the high precision of concentration on two-phase interfaces. Then, directional filtering algorithms and regularized-finite-difference (rLBM-FD) collision operator were introduced to improve the numerical stability and limit the numerical diffusion. Four common collision operators were implemented and thoroughly tested in two cases to verify the robustness and accuracy of our new model. In conclusion, the combination of the rLBM-FD with standard non-linear filter reaches the highest robustness, mass-conservativeness, and limitation on numerical diffusion. The directional non-linear filter has the lowest computational cost of any microscopic variable filter and can increase the robustness by two times. Macro-variable filtering is not appropriate for treating the two-film equilibrium because the mass loss and robustness are unacceptable.

Nonlinear trajectory estimation using extended state-space recursive least squares.

In a variety of filtering applications including target tracking, global positioning systems, and autonomous robots, the nonlinear nature of the system model makes estimation tasks challenging. This study presents an extended state-space recursive least squares (ESSRLS) filter for nonlinear trajectory estimation. The paper focuses on ESSRLS filter derivation for non-autonomous systems and investigates its performance against the extended Kalman filter and unscented Kalman filter for maneuvering aircraft trajectory estimation applications. The major accomplishment of the proposed approach is nonlinear filtering, independent of a priori information about noise statistics and the provision of the tuning parameter (forgetting factor). This makes the ESSRLS a more suited candidate for practical nonlinear filtering applications. The simulation results show that in the presence of model uncertainties, data outages (occlusion), and large initial condition deviations, the proposed method gives superior estimation performance as compared to the state-of-the-art.

Fuzzy EKF-Based Intrusion Detection and Accurate State Estimation of Interconnected DC MGs With CPLs

With the advancement of digital technologies in complex and large-scale power systems, the issue of cyber-attack detection has become of paramount importance to maintain system reliability and performance. In this regard, accurate state estimation plays a key role and leads to proper supervisory decision-making. To detect cyber-attacks in direct current microgrids (DC MGs), a new centralized attack detection system based on the Kalman filter and clustering approach is proposed. On the other hand, appearing nonlinear loads such as constant power loads (CPLs) make the overall system behavior nonlinear and conventional linear Kalman filters ineffective. So, in this paper, an extended Kalman filter (EKF) is utilized. By constructing several output vectors and parallel EKFs, local state estimations are achieved. They are then used to detect attacks based on the residuals and localize them based on the fuzzy clustering approach. The local healthy estimations are aggregated to compute global estimations with low noise. The simulation results on a five-area interconnected DC MG subjected to data integrity and denial-of-service (DoS) attacks show that the proposed approach can detect attacks, evaluate how many attacks occur in the case of multiple attacks, localize the attacks, and obtain accurate state estimations.

A study on the implementation of nonlinear Kalman filter applying MMG model

Many technologies need to be established to realize autonomous ships. In particular, accurate state estimation in real time is one of the most important technologies. In the ship and ocean engineering fields, there have been many studies on state estimation using nonlinear Kalman filters. Several methods have been proposed for nonlinear Kalman filters. However, there is insufficient verification on the selection of which filter should be applied among them. Therefore, this study aims to validate the filter selection to provide a guideline for filter selection. The effects of modeling error, observation noise, and type of maneuvers on the estimation accuracy of the unscented Kalman filter (UKF) and ensemble Kalman filter (EnKF) used in this study were investigated. In addition, it was verified whether filtering could be performed in real time. The results show that modeling error significantly impacts the estimation accuracy of the UKF and EnKF. However, the observation noise and types of maneuvers did not have an impact like the modeling error. Thus, we obtained the guideline that UKF and EnKF should be used differently depending on the required computation time. We also obtained that keeping the modeling error sufficiently small is essential to improving the estimation accuracy.

Development and Dynamic State Estimation for Robotic Knee–Ankle Orthosis With Shape Memory Alloy Actuators

Abstract The development of rehabilitation robots has long been an issue of increasing interest in a wide range of fields. An important aspect of the ongoing research field is applying flexible components to rehabilitation equipment to enhance human−machine interaction. Another major challenge is to accurately estimate the individual’s intention to achieve safe operation and efficient training. In this article, a robotic knee−ankle orthosis (KAO) with shape memory alloy (SMA) actuators is developed, and the estimation method is proposed to determine the joint torque. First, based on the analysis of human lower limb structure and walking patterns, the mechanical design of the KAO that can achieve various rehabilitation training modes is detailed. Next, the dynamic model of the hybrid-driven KAO is established using the thermodynamic constitutive equation and Lagrange formalism. In addition, the joint torque estimation is realized by the nonlinear Kalman filter method. Finally, the prototype and human subject experiments are conducted, and the experimental results demonstrate that the KAO can assist lower limb movements. In the three experimental scenarios, reductions of 59.1%, 16.5%, and 73% of the torque estimation error during the knee joint movement are observed, respectively.

A Novel Adaptive Swallow Swarm Optimisation Method for Medical Image Segmentation in Brain MRI Using Machine Learning

Automatic segmentation has to be effective to provide sufficient results for clinical use due to inherent challenges involved in medical images, including poor image quality, different imaging, segmentation protocols or variations in patients. Further, they are reduced to previously invisible classes by a lack of image-specific adaptation and the absence of generality. When handling some complex cases, it may lead to ineffectiveness, taking into account the differences in distribution between training and test data. To solve these issues in the proposed work, noise removal and segmentation algorithm is introduced by using Adaptive Swallow Swarm Optimization (ASSO) with Convolutional Neural Network (CNN) and is called as ASSO-CNNseg to find traces of cancer in Brain Tumour images. In the noise removal step nonlinear filtering algorithm is introduced to remove noises from the image samples efficiently. After the removal of noises from the image samples, then segmentation is performed by using the deep learning algorithm consists of some steps: image-based fine tuning, interaction-oriented uncertainty, and segmentation process. In the fine-tuning algorithm, ASSO algorithm is introduced for weighted loss feature taking the network and interaction-based uncertainty taken into account during image-dependent fine-tuning. Next, here the initial effort make use of CNNseg to tackle their earlier unknown objects are the image segmentation context is presented. The experimental outcomes demonstrate the highly accurate results of the proposed ASSO-CNNseg method with less number interactions between users and less consumption of time for user compared to classical interactive segmentation techniques such as Slic-seg and BIFSeg.