Kalman Estimator Research Articles

Financial stability influence on climate risk, GHG emission, and green economic recovery of China.

This study examines the nexus between financial stability, climate risks, GHG emission mitigation, and green economic recovery of China. Financing efforts to protect against and reduce the hazards associated with climate change need to consider these risks and resources. Study used the Kalman technique of analysis for empirical inference. This research focuses on the carbon risk in China by employing a Kalman estimation approach. Although environmental mitigation was found to be important at 39%, financial strength and carbon hazards were considerable at 34%. Moreover, the report demonstrates the relationship between climatic threats and environmental drift in China, at a rate of 17%, emphasizing the need to address climate change issues. A state's fiscal health guarantees national economic security while pursuing green economic recovery initiatives. Researchers concluded that precise policy suggestions were needed to promote green economic development.

Robust weighted fusion Kalman estimators for systems with uncertain noise variances, multiplicative noises, missing measurements, packets dropouts and two‐step random measurement delays

AbstractFor multisensor networked systems with uncertain noise variances, multiplicative noises and multiple networked‐induced uncertainties including missing measurements, packets dropouts and two‐step random measurement delays, the robust weighted fusion estimation problem is addressed in this article. More precisely, the system noise variances are assumed to be uncertain but bounded, the other four uncertainties are compensated into fictitious white noise by the proposed model transformation method, which includes the augmented method and extended fictitious noise technique. Then local multi‐model system is obtained, for which robust local Kalman estimator is obtained based on the minimax robust estimation principle and unified estimation method. Based on this, the six robust weighted fusion time‐varying Kalman estimators are presented in a unified form, which include robust weighted fusers weighted by matrices, diagonal, scalars, and a robust covariance intersection (CI) fuser and two fast CI (FCI) fusers. The robustness proving method, including the extended Lyapunov equation approach with two kinds of generalized Lyapunov equations, non‐negative matrix factorization and elementary transformation of matrix, is presented to prove that the actual estimation error variances are guaranteed to have minimal upper bounds for all admissible uncertainties. The accuracy relations are proved. Further, the robust local and fused steady‐state Kalman estimators are presented. Finally, a simulation example applied to Internet‐based three tank water system is given to demonstrate effectiveness of the proposed results.

Flow Reconstruction Around a Surface-Mounted Prism from Sparse Velocity and/or Scalar Measurements Using a Combination of POD and a Data-Driven Estimator

A data-driven algorithm is proposed for flow reconstruction from sparse velocity and/or scalar measurements. The algorithm is applied to the flow around a two-dimensional, wall-mounted, square prism. To reduce the problem dimensionality, snapshots of flow and scalar fields are processed to derive POD modes and their time coefficients. Then a system identification algorithm is employed to build a reduced order, linear, dynamical system for the flow and scalar dynamics. Optimal estimation theory is subsequently applied to derive a Kalman estimator to predict the time coefficients of the POD modes from sparse measurements. Analysis of the flow and scalar spectra demonstrate that the flow field leaves its footprint on the scalar, thus extracting velocity from scalar concentration measurements is meaningful. The results show that remarkably good reconstruction of the flow statistics (Reynolds stresses) and instantaneous flow patterns can be obtained using a very small number of sensors (even a single scalar sensor yields very satisfactory results for the case considered). The Kalman estimator derived at one condition is able to reconstruct with acceptable accuracy the flow fields at two nearby off-design conditions. Further work is needed to assess the performance of the algorithm in more complex, three-dimensional, flows.

A Distributed Implementation of Steady-State Kalman Filter

This paper studies the distributed state estimation in sensor network, where <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> sensors are deployed to infer the <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> -dimensional state of a Linear Time-Invariant (LTI) Gaussian system. By a lossless decomposition of the optimal steady-state Kalman filter, we show that the problem of distributed estimation can be reformulated as that of the synchronization of homogeneous linear systems. Based on such decomposition, a distributed estimator is proposed, where each sensor node runs a local filter using only its own measurement, alongside with a consensus algorithm to fuse the local estimate of every node. We prove that the average of estimates from all sensors coincides with the optimal Kalman estimate, and under certain condition on the graph Laplacian matrix and the system matrix, the consensus error is bounded and the asymptotic error covariance is derived. As a result, the distributed estimator is stable for each single node. We further show that the proposed algorithm has a low message complexity of min( <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$m$</tex></formula> , <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$n$</tex></formula> n). Numerical examples are provided in the end to illustrate the efficiency of the proposed algorithm.

Using a population-based Kalman estimator to model the COVID-19 epidemic in France: estimating associations between disease transmission and non-pharmaceutical interventions.

In response to the COVID-19 pandemic caused by SARS-CoV-2, governments have adopted a wide range of non-pharmaceutical interventions (NPI). These include stringent measures such as strict lockdowns, closing schools, bars and restaurants, curfews, and barrier gestures such as mask-wearing and social distancing. Deciphering the effectiveness of each NPI is critical to responding to future waves and outbreaks. To this end, we first develop a dynamic model of the French COVID-19 epidemics over a one-year period. We rely on a global extended Susceptible-Infectious-Recovered (SIR) mechanistic model of infection that includes a dynamic transmission rate over time. Multilevel data across French regions are integrated using random effects on the parameters of the mechanistic model, boosting statistical power by multiplying integrated observation series. We estimate the parameters using a new population-based statistical approach based on a Kalman filter, used for the first time in analysing real-world data. We then fit the estimated time-varying transmission rate using a regression model that depends on the NPIs while accounting for vaccination coverage, the occurrence of variants of concern (VoC), and seasonal weather conditions. We show that all NPIs considered have an independent significant association with transmission rates. In addition, we show a strong association between weather conditions that reduces transmission in summer, and we also estimate increased transmissibility of VoC.

Long Short-Term Memory-Based Neural Networks for Missile Maneuvers Trajectories Prediction⋆

Due to its extensive applications in different contexts, moving target tracking has become a hot topic in the last years, above all in the military field. Specifically, missile tracking research received a great effort, mainly for its importance in terms of security and safety. Herein, traditional solutions, e.g. Interacting Multiple Model (IMM) based on the Kalman estimation theory, achieve good performance under the main restrictive assumption of the a priori knowledge of the target model, so neglecting the unavoidable presence of model uncertainties and limiting the achievable tracking accuracy only by the presence of the measurement noise. With the specific aim of overcoming this narrowness, this work investigates the capability of deep neural networks in predicting the missile maneuvering trajectories in a model-free fashion. The idea is to leverage the Long-Short Term Memory (LSTM) net due to its excellent capability in learning long-term dependencies of temporal information. Two different LSTM-based architectures have been hence designed to predict both position and velocity of a missile using raw and noisy measurements provided by a realistic radar system, exploiting a large database abundant of realistic off-line data. Training results and theoretical derivations are verified through non-trivial scenarios in order to assess the capability of predicting unknown and realistic 3D missile maneuvers. Finally, the proposed approach has been also compared with a performing model-based IMM algorithm, suitably tuned to deal with realistic missile maneuvers, confirming the excellent generalization abilities of the developed data-driven architectures for different datasets.

Multi-pass Extended Kalman Smoother with Partially-known Constraints for Estimation of Vapor Compression Cycles

State and parameter estimation methodologies have the potential to make a significant impact in the development of broad array of capabilities for widely-used vapor compression cycles, including advanced controls, performance monitoring, data-driven modeling, and deployment of digital twin technologies. However, the nonlinearity and numerical stiffness of large physics-based models of these systems pose challenges for the practical implementation of estimators that must also satisfy the physical state constraints. We present a three-pass fixed-interval smoothing method developed in the extended Kalman estimation formalism that incorporates linear inequality and partially-known nonlinear equality constraints defined in terms of unknown parameters of the system. The smoothing method is demonstrated to have high estimation accuracy during joint state and parameter estimation of the cycle model representing a realistic system that is implemented in Julia language leveraging automatic differentiation capabilities.

M-estimation based Robust Approach for Hybrid Dynamic State Estimation in Power Systems

Background: The state estimation (SE) process in power systems estimates bus voltage magnitude and phase angles vital for operating the system securely and reliably. The power systems state estimation problem has been extensively solved through a weighted least squares (WLS) based static approach that fails to track the system dynamics. Furthermore, those approaches are not inherently robust against outliers, yielding a separate bad data processing (BDP) technique. Popular Dynamic state estimation (DSE) schemes which mainly employ nonlinear Kalman filters like Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), also suffer from providing a reasonable estimation of states in the presence of bad data. Although several generalized maximum (GM) likelihood- based DSE approaches are robust against outliers, they are mainly based on nonlinear Kalman filters, which yield an iterative process. Therefore, this article focuses on developing a robust DSE approach that gives a good estimation of states against outliers in a single iteration. Objective: This article aims to propose a robust hybrid non-iterative DSE approach that gives robust SE results in the presence of bad data. Methods: The proposed novel robust hybrid DSE (NRHDSE) approach combines the robust Mestimation with the original novel hybrid DSE (NHDSE) approach. The proposed scheme implements a suitable linear relationship between integrated hybrid measurements and complex states. The proposed method uses a linear measurement model and thus employs an optimal linear Kalman filter to correct or estimate states. Results: The efficacy of the proposed approach has been demonstrated by applying it on IEEE 57, 118 bus test systems and one more extensive 246 Indian utility bus system, namely, Northern Regional Power Grid (NRPG), and after that comparing it with the original NHDSE, and DSE methods based on traditional EKF and M-estimation based robust version of EKF (REKF). The simulation result demonstrates the superiority of the proposed approach. Conclusions: Obtained results clearly show the superiority of the proposed approach.

Fusion steady‐state robust filtering for uncertain multisensor networked systems with application to autoregressive moving average signal estimates

AbstractThis article is concerned with the centralized fusion (CF) robust steady‐state Kalman filtering problem for a class of multisensor networked systems with mixed uncertainties including state‐dependent and noise‐dependent multiplicative noises, missing measurements, packet dropouts, and uncertain noise variances. By using a model transformation approach, the initial multisensor system under study is transformed into a multi‐model multisensor system only with uncertain noise variances. By introducing an augmented state vector, the CF system is obtained. In the light of the minimax robust estimation method, and based on the worst‐case fusion system with conservative upper bounds of uncertain noise variances, the CF robust steady‐state Kalman estimators (predictor, filter, and smoother) are proposed in a unified form. By means of the permutation matrices and Lyapunov equation method, the robustness of CF estimators is proved. The accuracy relations among the robust local and fusion steady‐state estimators are proved. An example applied to autoregressive moving average signal estimation is put forward, and the robust CF steady‐state signal estimators are proposed. Simulation experiment shows the correctness of the proposed approach.

Estimating gravity acceleration from an atomic gravimeter by Kalman filtering

We present the construction of a two-state model of the atomic gravimeter and the associated Kalman recursion to estimate gravity acceleration from an atomic gravimeter. It is found that the Kalman estimator greatly improves the estimation precision in the short term by removing the white phase noise. The residual noise of the estimates follows for more than and highlights a precision of at the measuring time of a single sample, even with no seismometer correction.

Decentralized X-ray pulsar-based navigation system for space-borne gravitational wave detector

In this paper, the inter-satellite link (ISL) aided X-ray pulsar-based navigation (XPNAV) system is studied to improve the survivability of space-borne gravitational wave detector. An unscented transform (UT) based distributed Kalman estimator is proposed for state estimation within less computational and communication costs. Three different navigation scenarios are designed based on a typical large-scale distributed heliocentric orbital space-borne gravitational wave detector, Laser Interferometer Space Antenna (LISA). Simulation results show that the positioning root-mean-square error (RMSE) reduces ̃28% after introducing the ISL to XPNAV. Moreover, the proposed distributed Kalman estimator and the classical Kalman estimator shows similar performance in XPNAV/ISL integrated navigation system with the positioning RMSE of ̃528m and ̃520m, respectively. Compared with the classical method, both analytical and simulating analyses demonstrate that the proposed algorithm efficiently reduces the computational costs to 1/N2 for a team with N agents and avoids additional communication needs simultaneously. The efficient distributed Kalman estimator-based XPNAV/ISL integrated navigation provides a high-performance autonomous navigation solution for interplanetary spacecraft formation flying missions.

GA-ELM-based adaptive Kalman estimator for SOC of lithium-ion batteries

GA-ELM-based adaptive Kalman estimator for SOC of lithium-ion batteries

Estimation of Liquid Level in a Harsh Environment Using Chaotic Observer

The increased demand for liquid level measurement has been a key factor in designing accurate and reliable control systems. Here, a study was carried out to calculate the liquid level in a tank using a pressure sensor for changes in inlet liquid parameters like temperature, density and velocity. Prediction of their variables for the long term is essential due to the randomness present in the input and measurement. Hence, observer design for state estimation of a non-linear dynamic system with uncertainties in the measurement and process becomes important. This work provides a feedback observer solution for a system with multiple inputs and single measurable output. A full state observer model is developed to estimate a system’s states with a sensor placed at a definite position from the pipe’s input point through which the liquid flows at different densities and temperatures. Using the observability properties, Luenberger full state observer is designed by various methods, verified using MATLAB and SIMULINK for the system state estimation. To incorporate process noise and measurement noise, the Kalman estimator is integrated with the system. Chaotic systems are susceptible to initial conditions, variations in parameters and are complex dynamic systems. However, providing consistently precise measurements through particular meters necessitates time-consuming computations that can be reduced by employing machine learning approaches that make use of optimizers. The results obtained are compared with the prediction models obtained using Artificial Neural Networks and are validated through the readings obtained from the experimental setup.

Efficient secure state estimation against sparse integrity attack for regular linear system

AbstractWe consider the problem of estimating the state of a time‐invariant linear Gaussian system in the presence of integrity attacks. The attacker can compromise out of sensors, the set of which is fixed over time and unknown to the system operator, and manipulate the measurements arbitrarily. Under the assumption that the system is regular and system matrix is non‐singular, we propose a secure estimation scheme that is resilient to ‐sparse attack as long as the system is ‐sparse detectable, which achieves the fundamental limit of secure dynamic estimation. In the absence of attack, the proposed estimation coincides with Kalman estimation with a certain probability that can be adjusted to trade‐off between performance with and without attack. Furthermore, the detectability condition checking in the designing phase and the estimation computing in the online operating phase are both computationally efficient. Two numerical examples including the IEEE 68 bus test system are provided to corroborate the results and illustrate the performance of the proposed estimator.

Performance evaluation of neural network topologies for online state estimation and fault detection in pressurized water reactor

The nuclear reactor is a multi-rate nonlinear system in which the state variables progress with widely varying dynamics. It has state variables such as reactivity and delayed neutron precursor densities that cannot be measured directly via sensors. Reactivity signifies the criticality of the reactor core. Delayed neutron precursors are the source for delayed neutrons which plays a vital role in the change of neutron densities. Besides, the other states which are measured are also corrupted by measurement noise and are susceptible to sensor faults. Thus, estimation of these state variables becomes critical. As traditional estimators like EKF, UKF, and Particle filers require a close model of the process, Data-driven Neural networks architectures like Feed-Forward Neural networks, Dynamic NARX Neural networks, and Recurrent Neural networks are designed to estimate the reactor core states. The performance of the selected neural estimator is also compared with the Unscented Kalman estimator.

Centralized fusion robust filtering for networked uncertain systems with colored noises, one-step random delay, and packet dropouts

This paper studies the estimation problem for multisensor networked systems with mixed uncertainties, which include colored noises, same multiplicative noises in system parameter matrices, uncertain noise variances, as well as the one-step random delay (OSRD) and packet dropouts (PDs). This study utilizes the centralized fusion (CF) algorithm to combing all information received by each sensor, which improve the accuracy of the estimation. By using the augmentation method, de-randomization method and fictitious noise techniques, the original uncertain system is transformed into an augment model with only uncertain noise variances. Then, for all uncertainties within the allowable range, the robust CF steady-state Kalman estimators (predictor, filter, and smoother) are presented based on the worst-case CF system, in light of the minimax robust estimation principle. To demonstrate the robustness of the proposed CF estimators, the non-negative definite matrix decomposition method and Lyapunov equation approach are employed. It is proved that the robust accuracy of CF estimator is higher than that of each local estimator. Finally, the simulation example applied to the uninterruptible power system (UPS) with colored noises and multiple uncertainties illustrates the effectiveness of the proposed CF robust estimation algorithm.

Position Estimation and Error Correction of Mobile Robots Based on UWB and Multisensors

Since there are many interferences in the indoor environment, it is difficult to achieve the precise positioning of the mobile robot using a single sensor. This paper presents a position estimation and positioning error correction method of mobile robots based on multisensor data. The robot’s positioning sensor includes ultra-wideband (UWB) components, inertial measurement unit (IMU), and encoders. UWB multipath interference causes more ranging errors, which can be reduced by the correction equation after data fitting. The real-time coordinates of the UWB robot tag can be calculated based on multiple UWB anchor data and the least squares method. The coordinate data x c , y c are acquired by UWB positioning subsystem, and the velocity data x ̇ c , y ̇ c are collected by IMU together with encoders. The multisensor data continuously update Kalman filter and estimate robot position. In the positioning process, the positioning data of different sensors can be mutually corrected and supplemented. The results of UWB ranging correction experiments indicate that data fitting can improve the UWB positioning accuracy. In the multisensor positioning experiments, compared with a single sensor, the positioning method based on data fusion of UWB, IMU, and encoders has higher accuracy and adaptability. When UWB signals are interfered or invalid, other sensors can still work normally and complete the robot positioning process. The multisensor positioning method not only improves the robot positioning accuracy but also has stronger environmental adaptability.

Experimental evaluation for fluxes, currents and speed estimation of induction motor

Induction motor drive control and estimation is a wide subject. The market for variable speed drives has grown dramatically in the last few years. Manufacturers have recognized the importance of not only managing the speed or torque range, but also reducing power consumption. This necessitates the development of new control algorithms and schemes to include these solutions. Indeed, the speed estimate must be employed in one or more regions of the control scheme, depending on the control objective. This concept, as well as the most common speed estimation methodologies, is investigated. Currently, many tools can be used for the evaluation of the rotor speed without a speed sensor. By modern signal processing methods, it is possible to implement an estimation scheme with the possibility of monitoring currents and voltages. Therefore, in this paper, the concept of currents, speed and fluxes estimation based on the extended Kalman filter is proposed. By monitoring the ratio of the theoretical residual to the actual residual, the measured noise covariance matrix is recursively corrected online to make it gradually approach the real noise level. So that the filter performs the optimal estimation, improves the accuracy of the speed estimation. The effect of the load change on the currents, fluxes and speed estimation was also studied. Simulation and experimental results show that the proposed improved adaptive extended Kalman estimator has a strong ability to suppress random measurement noise. The experimental and simulation results prove the accuracy of the proposed scheme towards the state estimation of an induction motor at different load levels. It can accurately estimate the speed of the motor and has a good anti-error ability to meet the actual needs of the project.

Control of Drop Volume and Drop Jetting Velocity in Inkjet Printing

Consistent dosages with high deposition accuracy are critical for drop-on-demand(DoD) inkjet printing to be used in additive manufacturing and device characterization. Practically, drop volume and drop jetting velocity are subject to process uncertainties, such as applied pressure fluctuation and variation in printheads, for which open-loop approaches are unable to compensate. In this work, a stochastic system model of the relation between two control parameters of the firing waveform and two outputs, drop volume and drop jetting velocity, is developed from experimental data. An image-based control strategy which comprises a one-step look ahead controller combined with a Kalman estimator for model parameters estimation is proposed to control the drop volume and drop jetting velocity. The effectiveness of the proposed control strategy is validated experimentally.

Links between Kalman Filtering and Data Assimilation with Generalized Least Squares

Kalman filtering (KF) is a popular form of data assimilation, especially in real-time applications. It combines observations with an equation that describes the dynamic evolution of a system to produce an estimate of its present-time state. Although KF does not use future information in producing an estimate of the state vector, later reanalysis of the archival data set can produce an improved estimate, in which all data, past, present and future, contribute. We examine the case in which the reanalysis is performed using generalized least squares (GLS), and establish the relationship between the real-time Kalman estimate and the GLS reanalysis. We show that the KF solution at a given time is equal to the GLS solution that one would obtain if data excluded future times. Furthermore, we show that the recursive procedure in KF is exactly equivalent to the solution of the GLS problem via Thomas’ algorithm for solving the block-tridiagonal matrix that arises in the reanalysis problem. This connection suggests that GLS reanalysis is better considered the final step of a single process, rather than a “different method” arbitrarily being applied, post factor. The connection also allows the concept of resolution, so important in other areas of inverse theory, to be applied to KF formulations. In an exemplary thermal diffusion problem, model resolution is found to be somewhat localized in both time and space, but with an extremely rough averaging kernel.