Recursive Estimation Research Articles

ON THE CONVERGENCE OF RECURSIVE KERNEL DENSITY ESTIMATORS FOR WIDELY ORTHANT DEPENDENT AND CENSORED DATA

In this article, we establish the almost sure convergence of a family of recursive estimators when the data are censored checking widely orthant dependence (WOD). The dependence hypothesis gives this work a certain originality because most often, the study of censored data is done with an independence hypothesis and in reality the real data are often dependent.

Asymptotic properties of recursive kernel density estimation for long-span high-frequency data

. This article mainly studies the asymptotic properties of recursive kernel density estimation for high-frequency data with a long time span. Under appropriate conditions, the variance, bias, mean squared error, and optimal bandwidth are obtained. The asymptotic normality is proven by using moment inequalities for ρ-mixing high-frequency data. Numerical simulations show that the recursive kernel density estimation provides good fitting results for the invariant density function of the Vasicek diffusion process. Empirical analysis reveals that recursive kernel density estimation not only reflects the distribution characteristics of financial data but also allows us to fit parameter models by minimizing the squared deviation between the recursive kernel density estimation and the parameter model. It provides more applications for the financial field.

Recursive dynamic state estimation for power systems with an incomplete nonlinear DAE model

AbstractPower systems are highly complex, large‐scale engineering systems subject to many uncertainties, which makes accurate mathematical modeling challenging. This article introduces a novel centralized dynamic state estimator designed specifically for power systems where some component models are missing. Including the available dynamic evolution equations, algebraic network equations, and phasor measurements, the least squares criterion is applied to estimate all dynamic and algebraic states recursively. The approach generalizes the iterated extended Kalman filter and does not require static network observability, relying on the network topology and parameters. Furthermore, a topological criterion is established for placing phasor measurement units (PMUs), termed topological estimability, which guarantees the uniqueness of the solution. A numerical study evaluates the performance under short circuits in the network and load changes and shows superior tracking performance compared to robust procedures from the literature with computational times in accordance with the typical PMU sampling rates.

Solar powered integrated multi sensors to monitor inland lake water quality using statistical data fusion technique with Kalman filter

This study proposes a data-driven statistical model using multi sensor fusion and Kalman filtering for real-time water quality assessment in lakes. A recursive estimation technique, the Kalman Filter, is employed to handle uncertainties and enhance computational efficiency. The fusion process integrates data from sensors monitoring parameters like chlorophyll concentration, surface water elevation, temperature, and precipitation, producing Markov features to capture temporal transitions and environmental dynamics. Data synchronization and fusion are achieved through recursive KF methods, enabling real-time adaptive management in response to environmental fluctuations such as seasonal changes, precipitation (6–18%), and evaporation rates (1.2–11.9 mm/day). Over a 30-day evaluation period, the model accurately predicted chlorophyll concentrations, reaching 128 mg/m3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {mg}/\\hbox {m}^{3}$$\\end{document} in mid-level inflow regions (3.6 m water elevation) compared to 86 mg/m3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {mg}/\\hbox {m}^{3}$$\\end{document} in extreme inflow areas (5.5 m). The integration of Markov feature extraction and eigenvalue estimation enhanced prediction stability and sensitivity, with the KF maintaining computational efficiency at 7.8 ms per computation cycle. The model’s accuracy was validated by achieving a residual error of less than 0.05 with minimal noise interference. Overall, the system provides a resilient and precise framework for real-time lake water quality assessment, capable of handling multi-parameter uncertainties and dynamic environmental changes, thereby supporting informed decision-making for aquatic ecosystem management.

Recursive state estimation for delayed complex networks with random link failures and stochastic inner coupling under cyber attacks

This paper focuses on the filtering issue regarding a class of uncertain time-delayed complex networks (CNs) subject to random link failures (RLF) and stochastic inner coupling (SIC) under cyber attacks, where the attacks occur in a random way with uncertain occurrence probabilities. The phenomena of RLF and SIC are described via a Bernoulli distributed random variable and a Gaussian noise, respectively. The former reflects the presence or absence of connections between different nodes, while the latter presents the uncertainty of the inner coupling strength. In addition, denial of service attacks (DoSAs) and false data injection attacks (FDIAs) are both taken into account in the discussed cyber space, where the switching behavior of attack modes is described by two independent Bernoulli distributed random variables. A novel recursive estimator is constructed with consideration of RLF, uncertain time-delay, SIC and cyber attacks, wherein a state estimation error covariance upper bound (SEECUB) is obtained. And such upper bound is minimized via reasonable design of the estimator gain. Through theoretical deduction and analysis, it is proved that the presented SEECUB is uniformly bounded under certain conditions. Finally, the availability of the proposed filtering strategy and the gained results are verified through simulation examples.

Fine-tuning weibull distribution parameters in Morocco’s Tarfaya and Tangier wind farms using two-stage swarm optimization

This study assesses the efficacy of particle swarm optimization (PSO) in estimating scale ( c) and shape ( k) parameters of the Weibull distribution model for wind energy forecasting at two key wind farm sites in Morocco— Tarfaya in the south and Tangier in the north, utilizing real wind data from 2022. Employing a novel square frequency error objective function to enhance parameter accuracy, the study adopts a two-stage training approach involving recursive least-square estimation and PSO fine-tuning. Validation with artificial data underscores PSO’s effectiveness under diverse wind conditions. Parameter sensitivity analysis identifies four optimal PSO configurations, with the PSO-4 model exhibiting superior performance. Comparative analysis against traditional and heuristic optimization methods consistently demonstrates PSO-4’s lowest root-mean-square error (RMSE) and mean absolute error (MAE), high coefficients of determination ( R2), and shortest computation time. The research highlights PSO-4 model as a precise and efficient tool for Weibull distribution parameter estimation in wind energy forecasting, showcasing robust convergence across both wind farm sites.

Model predictive control based integrated fault tolerant controller for autonomous vehicles with four-wheel independent steering, driving and braking systems

This paper presents a fault-tolerant control algorithm for path and velocity tracking of an autonomous vehicle based on four-wheel independent steering, driving, and braking systems. The proposed algorithm is designed to distribute the control effort to normal actuators in the occurrence of faults. The proposed algorithm consists of three modules. The reference velocity and yaw rate to track the reference path are determined by the reference state decision module. The model predictive control (MPC)-based fault-tolerant controller determines the optimal inputs to compensate for the effect of the faulty actuator and maintain the tracking performance. A faulty-factor estimator is proposed based on a recursive covariance estimator to evaluate the performance degradation due to the failure of each actuator. The estimated faulty factors are applied to the objective function of the MPC such that the control input is distributed based on the level of the fault. The performance of the proposed algorithm was verified via performing simulations using MATLAB/Simulink and CarSim. The simulations proved that the proposed controller maintained the control performance of the normal case even in multiple actuator failures.

Adaptive Model Predictive Control of Yaw Rate and Lateral Acceleration for an Active Steering Vehicle Based on Tire Stiffness Estimation

Dynamic systems for vehicles are commonly established based on physical rules that are simplified and are not accurately reflective of their dynamic characteristics under certain operating conditions, affecting their accuracy and safety. This paper presents an adaptive model predictive controller (AMPC) with an estimator that controls the yaw rate and lateral acceleration of a realistic ground vehicle. A number of vehicle parameters are estimated using different advanced estimators, of which recursive least squares is one that is widely used. AMPCs and estimators are used to manage the driving process in order for vehicles to remain stable and controllable. In this research, experiments were conducted on a realistic vehicle based on a nonlinear brush tire model. In this estimator, lateral force measurements are analyzed to estimate the tire cornering stiffnesses that are used in the AMPC from a linear bicycle model (lateral force model), where each parameter describes the nonlinearity of the vehicle model. The results demonstrate that the controlled vehicle's performance is improved by combining a recursive least squares estimator with an AMPC in the simulation process. As tire stiffness estimates become more accurate, AMPC performance improves. Yet, AMPC controllers are described in terms of a table of design parameters. As different steering inputs are applied to different vehicles, the yaw rate and lateral acceleration are varied, while tire stiffness is determined. According to the results of this paper, the proposed method for estimating tire-cornering stiffness exhibits high estimation accuracy, robustness, computational efficiency, stability, comfort, and accuracy and reliability under a variety of steering input maneuvers using an AMPC controller.

Reduced-order reconstruction of discrete grey forecasting model and its application

Discrete grey forecasting models based on an accumulative operator have been widely used in many practical fields. With the development of grey forecasting models, it is a problem to be solved to further analyze internal mechanisms and unify the structures. This paper aims to reconstruct the model from a perspective of sequence characteristics and simplify the modeling steps under the condition of ensuring the accuracy of the model. First, this paper analyzes dynamic sequence evolution hidden and mines relationship between the structure and original sequence features contained in discrete grey forecasting model. Then, the reconstruction is carried out to prove the equivalence and quantitative relation between reduced-order model and original model. Under order recursive estimation, new parameters are addressed. Finally, theoretical correctness is verified by large-scale numerical simulation. Moreover, the reduced-order model is applied for prediction on the peak of battery incremental capacity and capacity degradation. Results show the effectiveness and superior prediction performance of the reduced-order model, where MAPEs of grey forecasting models have controlled under 4%.

Adaptive model predictive control scheme for partially internal thermally coupled air separation column

Partially Internal Thermally Coupled Air Separation Column (P-ITCASC) is a highly energy-efficient and cost-effective technology. However, its complex dynamics resulting from thermal coupling pose a challenge to the operating stability of this technology. This article, therefore, proposes an adaptive model predictive control (AMPC) scheme for the P-ITCASC process. The controller incorporates an auto-regressive and exogenous inputs (ARX) model, a linear time-varying Kalman filter, and a recursive polynomial model estimator (RPME) algorithm. Within RPME, ARX polynomial models are identified and utilized to estimate the time-varying parameters and update the prediction model of the process. The process states are observed through the Kalman filter, and the constrained receding horizon optimization problem is solved using quadratic programming. This control scheme ensures the closed-loop system's feasibility and stability in the presence of output/input constraints. An adaptive generic model control, model predictive control, and adaptive internal model control schemes were also designed for benchmarking study. Numerical simulations show that AMPC is more efficient in handling nonlinear dynamics and maintaining product concentration to desired set points compared to other control schemes.

Optimal Inference of Hidden Markov Models Through Expert-Acquired Data.

This paper focuses on inferring a general class of hidden Markov models (HMMs) using data acquired from experts. Expert-acquired data contain decisions/actions made by humans/users for various objectives, such as navigation data reflecting drivers' behavior, cybersecurity data carrying defender decisions, and biological data containing the biologist's actions (e.g., interventions and experiments). Conventional inference methods rely on temporal changes in data without accounting for expert knowledge. This paper incorporates expert knowledge into the inference of HMMs by modeling expert behavior as an imperfect reinforcement learning agent. The proposed method optimally quantifies experts' perceptions about the system model, which, alongside the temporal changes in data, contributes to the inference process. The proposed inference method is derived through a combination of dynamic programming and optimal recursive Bayesian estimation. The applicability of this method is demonstrated to a wide range of inference criteria, such as maximum likelihood and maximum a posteriori. The performance of the proposed method is investigated through a comprehensive numerical experiment using a benchmark problem and biological networks.

Recursive and batch Bayesian estimation of exponential non-viscous damping systems

Non-viscous damping systems include a convolutional integral and a kernel function in their equation of motion to incorporate time-hysteresis damping effects, which are present in dynamical behavior of most engineering materials yet are absent in commonly used viscous damping models. When an exponential kernel function is adopted by the non-viscous model, the unique damping characteristic is realized by an additional model parameter known as the relaxation parameter. The objective of this paper is to investigate various aspects of estimation problems on exponential non-viscous damping systems through recursive and batch Bayesian time-domain frameworks, with a focus on how the relaxation parameter introduces additional complexity in estimation problems compared to viscous damping systems. Two well-established algorithms are employed: the unscented Kalman filter (UKF) for recursive Bayesian estimation and the transitional Markov Chain Monte Carlo (TMCMC) for batch Bayesian estimation. We first analyze the complexity of estimation problems in SDOF viscous and non-viscous damping systems. The TMCMC algorithm offers a quantitative analysis that reveals greater uncertainty and stronger parameter correlation in non-viscous damping systems compared to viscous damping systems. Subsequently, the same SDOF systems are utilized to study recursive Bayesian estimation via UKF such as the effects of measurement data type, discretization, and initial conditions. Based on the results of the TMCMC algorithm, we study the dependence of UKF performance on the initial condition for various SDOF non-viscous damping systems with different levels of non-viscosity. In the end, a laboratory experiment compares the estimation results between viscous and non-viscous damping systems using both TMCMC and UKF.

Noise-Adaptive State Estimators with Change-Point Detection.

Aiming at tracking sharply maneuvering targets, this paper develops novel variational adaptive state estimators for joint target state and process noise parameter estimation for a class of linear state-space models with abruptly changing parameters. By combining variational inference with change-point detection in an online Bayesian fashion, two adaptive estimators-a change-point-based adaptive Kalman filter (CPAKF) and a change-point-based adaptive Kalman smoother (CPAKS)-are proposed in a recursive detection and estimation process. In each iteration, the run-length probability of the current maneuver mode is first calculated, and then the joint posterior of the target state and process noise parameter conditioned on the run length is approximated by variational inference. Compared with existing variational noise-adaptive Kalman filters, the proposed methods are robust to initial iterative value settings, improving their capability of tracking sharply maneuvering targets. Meanwhile, the change-point detection divides the non-stationary time sequence into several stationary segments, allowing for an adaptive sliding length in the CPAKS method. The tracking performance of the proposed methods is investigated using both synthetic and real-world datasets of maneuvering targets.

An Improved Collaborative Estimation Method for Determining The SOC and SOH of Lithium-Ion Power Batteries for Electric Vehicles

With the increase in the amount of actual operating data on electric vehicles, how to analyze and process useful information from existing battery charging and discharging data and apply it to subsequent state estimation is worthy of in-depth thinking and practice by researchers. This article proposes a collaborative estimation architecture for SOC and SOH based on the 1RC equivalent circuit model, recursive least squares, and adaptive extended Kalman filtering algorithms (AEKF), which combine offline data processing with online applications. By applying offline data processing, OCV–SOC polynomial fitting and average polarization resistance were determined, which reduced the time required for basic data measurement and improved the accuracy of model parameter identification, while a recursive estimation combining micro- and macro-time-scales of AEKF was used for the online real-time estimation of the SOC and actual available capacity of batteries, in order to eliminate interference from measurement and process noise. The results of the simulated and experimental data validation indicate that the proposed algorithm is applicable to the lithium-ion batteries studied in this paper, the average SOC deviation is less than 1.5%, the maximum deviation is less than 2.02%, and the SOH estimation deviation is less than 1% under different driving conditions in the multi-temperature range. This study lays the foundation for further utilizing offline data and improving SOC and SOH collaborative estimation algorithms.

Parameter estimation methods for time‐invariant continuous‐time systems from dynamical discrete output responses based on the Laplace transforms

SummaryIn industrial process control systems, parameter estimation is crucial for controller design and model analysis. This article examines the issue of identifying parameters in continuous‐time models. This article presents a stochastic gradient estimation algorithm and a recursive least squares estimation algorithm for identifying the parameters of continuous systems. It derives the parameter identification model of linear continuous‐time systems based on the Laplace transforms of the input and output of the systems. To prove that the techniques given here work, we have included a simulated example.

Research and comparison of algorithms based on multi-modal fusion

Abstract A multi-modal fusion algorithm is an important method for information fusion on multi-modal data provided by different sensors. It can make full use of the advantages of multiple sensors and improve the accuracy and robustness of data processing and decision-making. This paper aims to study the performance difference between the extended Kalman filter (EKF) algorithm and other algorithms in multi-modal fusion and explore a method to fuse multiple algorithms further to improve the accuracy of fusion results. The author uses the classic test set data set for experiments to evaluate the performance of different algorithms. By comparing and analyzing the performance of each algorithm in data fusion tasks, we can get their advantages and disadvantages in different scenarios. Among them, the extended Kalman filter algorithm is a classical algorithm based on Bayesian filtering, which can estimate the system state through recursive state estimation and covariance update. In addition, the author also uses linear regression, random forest, and other algorithms to compare. Then, the author uses several test sets to evaluate the performance of each algorithm. Through a comprehensive analysis of the MSE and MAE errors output by the algorithm, the applicability, advantages, and disadvantages of each algorithm in different scenarios are obtained.

Exchange Rate Pass-Through to Domestic Prices: Is Inflation Targeting an Optimal Policy Choice for Pakistan?

This study investigates the effect of exchange rate dynamics and its impact on consumer prices in Pakistan. Using monthly data for the period 2004- 2022, the Recursive Vector Autoregressive (VAR) technique has been employed for the empirical analysis. The recursive VAR estimation technique yields impulse response functions that demonstrate the incompleteness of exchange rate pass-through (ERPT). The results of the VAR reflect that the consumer prices bring most of the error variance in itself for the period under consideration. The exchange rate has been established to be the second most significant source of variations in consumer prices. The empirical results suggest that the lower ERPT reduces the risk of exchange rate volatility. Hence the monetary authorities may take steps to adopt inflation targeting in Pakistan.

Modal Parameter Recursive Estimation of Concrete Arch Dams under Seismic Loading Using an Adaptive Recursive Subspace Method.

Modal parameter estimation is crucial in vibration-based damage detection and deserves increased attention and investigation. Concrete arch dams are prone to damage during severe seismic events, leading to alterations in their structural dynamic characteristics and modal parameters, which exhibit specific time-varying properties. This highlights the significance of investigating the evolution of their modal parameters and ensuring their accurate identification. To effectively accomplish the recursive estimation of modal parameters for arch dams, an adaptive recursive subspace (ARS) method with variable forgetting factors was proposed in this study. In the ARS method, the variable forgetting factors were adaptively updated by assessing the change rate of the spatial Euclidean distance of adjacent modal frequency identification values. A numerical simulation of a concrete arch dam under seismic loading was conducted by using ABAQUS software, in which a concrete damaged plasticity (CDP) model was used to simulate the dam body's constitutive relation, allowing for the assessment of damage development under seismic loading. Utilizing the dynamic responses obtained from the numerical simulation, the ARS method was implemented for the modal parameter recursive estimation of the arch dam. The identification results revealed a decreasing trend in the frequencies of the four initial modes of the arch dam: from an undamaged state characterized by frequencies of 0.910, 1.166, 1.871, and 2.161 Hz to values of 0.895, 1.134, 1.842, and 2.134 Hz, respectively. Concurrently, increases in the damping ratios of these modes were observed, transitioning from 4.44%, 4.28%, 5.42%, and 5.56% to 4.98%, 4.91%, 6.61%, and 6.85%%, respectively. The correlation of the identification results with damage progression validated the effectiveness of the ARS method. This study's outcomes have substantial theoretical and practical importance, facilitating the immediate comprehension of the dynamic characteristics and operational states of concrete arch dam structures.

Recursive state estimation for two-dimensional systems over decode-and-forward relay channels: A local minimum-variance approach

The paper is concerned with the recursive state estimation problem for a class of two-dimensional systems over decode-and-forward relay channels subject to packet dropouts. Taking into account the sensor's limited transmission ability, a decode-and-forward relay is deployed between the sensor and the remote estimator to enlarge the propagation distance and improve the communication quality. The packet dropout phenomenon is described by two sequences of mutually uncorrelated random variables that obey Bernoulli distributions. The main objective of this paper is to design a recursive state estimator for the underlying system with guaranteed estimation performance. An upper bound on the estimation error variance is firstly derived by resorting to some coupled difference equations, and then such an upper bound is locally optimized at each iteration through proper design of the estimator gain. Furthermore, the impacts of the packet dropouts and the decode-and-forward relay on the estimation performance are evaluated with rigorously theoretical analysis. Finally, an illustrative example is presented to validate the effectiveness of the proposed estimation scheme.

Parameter learning for Wiener systems with time‐delay state‐space model

AbstractThis paper discusses a novel scheme for learning the Wiener output error nonlinear system with time‐delay state‐space model. In the Wiener system, the dynamic linear block is approximated by time‐delay state‐space model, and the static nonlinear block is established using neural fuzzy network. Combined signals designed including separable signal and random signal are devoted to achieving parameters separation learning of the Wiener system, that is, the two blocks are learned independently. Firstly, using the properties of shift operator and transforming state‐space model with time‐delay into a representation with input and output, then linear dynamic block parameters are learned by the virtue of correlation analysis method in the condition of Gaussian signals. Moreover, a recursive extended least squares estimation is carried out to learn parameters of static nonlinear block and colored noise model under the condition of random signals. The efficiency and accuracy of proposed scheme are confirmed on experiment results of a numerical simulation and a typical practical nonlinear process, and experimental simulation results demonstrate that the learning scheme proposed obtains good learning precision.