Kalman-Bucy Filter Research Articles

Minimum-energy switching geometric filter on lie groups for differential-drive wheeled mobile robots

Accurate state estimation plays a critical role in various applications, such as tracking, regulation, and fault detection in robotic and mechanical systems. Typically, the Kalman–Bucy filter is used as a linear state observer for this purpose. However, real-world robots often exhibit complex behavior, characterized by a combination of dynamics, making it essential to employ hybrid filters. In this context, the Switching Kalman filter stands out as a well-established solution. In this article we aim to generalize the Brownian-Markov Stochastic Model, a hybrid dynamic model for differential-drive wheeled mobile robots, to the case of a mobile robot whose center of mass is not aligned to the wheels axle middle point, and to design a geometric hybrid state estimator by exploiting the Lie groups theory. The Brownian-Markov Stochastic Model features two modes: “grip” and “slip”. These modes correspond to ideal grip and lateral slippage, with transitions governed by a state-dependent Markov chain. To validate the proposed switching filter, we conduct MATLAB® simulations of the robot’s motion in a scenario prone to lateral grip loss, comparing the state estimates produced by the switching geometric filter with those obtained using the switching Kalman filter.

The gated cascade diffusion model: An integrated theory of decision making, motor preparation, and motor execution.

This article introduces an integrated and biologically inspired theory of decision making, motor preparation, and motor execution. The theory is formalized as an extension of the diffusion model, in which diffusive accumulated evidence from the decision-making process is continuously conveyed to motor areas of the brain that prepare the response, where it is smoothed by a mechanism that approximates a Kalman-Bucy filter. The resulting motor preparation variable is gated prior to reaching agonist muscles until it exceeds a particular level of activation. We tested this gated cascade diffusion model by continuously probing the electrical activity of the response agonists through electromyography in four choice tasks that span a variety of domains in cognitive sciences, namely motion perception, numerical cognition, recognition memory, and lexical knowledge. The model provided a good quantitative account of behavioral and electromyographic data and systematically outperformed previous models. This work represents an advance in the integration of processes involved in simple decisions and sheds new light on the interplay between decision and motor systems. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Kalman Filter for a Particular Class of Dynamic Object Images

We discuss the problem of estimating the state of a dynamic object by using observed images generated by an optical system. The work aims to implement a novel approach that would ensure improved accuracy of dynamic object tracking using a sequence of images. We utilize a vector model that describes the object image as a limited number of vertexes (reference points). Upon imaging, the object of interest is assumed to be retained at the center of each frame, so that the motion parameters can be considered as projections onto the axes of a coordinate system matched with the camera's optical axis. The novelty of the approach is that the observed parameters (the distance along the optical axis and angular attitude) of the object are calculated using the coordinates of specified points in the object images. For estimating the object condition, a Kalman-Bucy filter is constructed on the assumption that the dynamic object motion is described by a set of equations for the translational motion of the center of mass along the optical axis and variations in the angular attitude relative to the image plane. The efficiency of the proposed method is illustrated by an example of estimating the object's angular attitude.

Real-time estimation of vehicle inertia parameters based on Kalman-bucy filter

Vehicle inertial parameters such as mass and moments of inertia are required for most vehicle dynamic control systems. Due to the wide range variation of these parameters during vehicle operation, accurate estimation of their values in real-time plays an important role in improving the efficiency of vehicle control systems. In this article, the vehicle sprung mass and moments of inertia are estimated in real-time based on a Kalman-Bucy filter algorithm designed for a spatial vibration model of a two-axle truck. This proposed method requires measuring only the vertical, roll, and pitch velocity of the sprung mass and, therefore can reduce the sensor cost significantly. The simulation results for a random roughness road profile according to ISO 8608 class C with step variations in sprung mass and moments of inertia showed that the designed estimator rejected the process and measurement noises and tracked the real vehicle parameters effectively with acceptable errors

Volatility estimation of hidden Markov processes and adaptive filtration

The partially observed linear Gaussian system of stochastic differential equations with low noise in observations is considered. The coefficients of this system are supposed to depend on some unknown parameter. The problem of estimation of these parameters is considered and the possibility of the approximation of the filtering equations is discussed. An estimators are used for estimation of the quadratic variation of the derivative of the limit of the observed process. Then this estimator is used for nonparametric estimation of the integral of the square of volatility of unobservable component. This estimator is also used for construction of method of moments estimators in the case where the drift in observable component and the volatility of the state component depend on some unknown parameter. Then this method of moments estimator and Fisher-score device allow us to introduce the One-step MLE-process and adaptive Kalman–Bucy filter. The asymptotic efficiency of the proposed filter is discussed.

Filtering dynamical systems using observations of statistics.

We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density ρ(v,t) given noisy observations of the true density ρ†; this contrasts with the standard filtering problem based on observations of the state v. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities ρ. However, for the purposes of tractability, we seek algorithms in state space; specifically, we introduce a mean-field state-space model, and using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modeling.

Analysis of the ensemble Kalman–Bucy filter for correlated observation noise

Analysis of the ensemble Kalman–Bucy filter for correlated observation noise

Low-Rank Approximated Kalman-Bucy Filters Using Oja’s Principal Component Flow for Linear Time-Invariant Systems

Low-Rank Approximated Kalman-Bucy Filters Using Oja’s Principal Component Flow for Linear Time-Invariant Systems

Adaptive fuzzy sliding mode control of magnetic levitation system based on Interval Type-2 Fuzzy Neural Network Identification with an Extended Kalman–Bucy filter

The electromagnet levitation control system (EMS) is the core component of the magnetic levitation train. However, its nature is characterized by an inherent instability and strongly nonlinear dynamic. Additionally, modeling uncertainties and numerous exogenous perturbations make the design of the controller even more difficult for this system. To overcome these limitations, a Fuzzy Sliding Mode Control based on Interval Type-2 Fuzzy Neural Network Identification (FSMCF2NN) is designed for the EMS system. The suggested controller has a unique feature that allows the sliding surface and controller effort to be designed by using model-free techniques. As a result, the drawbacks of the conventional SMC schemes, such as the chattering effect, requiring prior knowledge of the system’s uncertainties bounds, and selection of the appropriate control gains, are eliminated. Also, the FSMCF2NN method, since it is designed based on intelligence methods not only minimizes the accumulation of state errors, but also generates the optimum control efforts of the EMS system to ensure stability in different situations. Furthermore, since the measurement of all EMS states is not achievable in reality, a qualified observer is needed to estimate unmeasured states. For this purpose, the Extended Kalman-Bucy filter (EKBF) is selected because of its superiority in managing disturbances for nonlinear systems. Furthermore, to confirm the robustness of the FSMCF2NN scheme in different scenarios, the results of the controller based on Linear Matrix Inequality (LMI), Linear Quadratic Regulation (LQR) controller, and the conventional Sliding Mode Controller (SMC) are compared with the proposed controller. Analyzing the results of simulation demonstrates that maintaining the levitated object is achievable with the proposed control scheme, effectively. At the same time, it attempts to adequately compensate for the uncertainties and disturbances present in the EMS structure.

An extended Kalman-Bucy filter for state of charge estimation of 2-RC network modelled Li-ion battery

Li-ion batteries have increased dramatically in the automobile industry over the past few decades. Lithium-ion batteries are viewed more favourably due to their high energy density, specific energy, etc. Optimising the performance of lithium-ion batteries requires accurate modelling and estimation of the state of charge (SOC). This article introduces an efficient second-order resistor-capacitor (2RC) network battery model and an optimised extended Kalman-Bucy filter (EKBF) as the most effective way to model a battery and estimate its state of charge (SOC). The operating parameters of a lithium-ion battery are measured with the least-squares curve fitting method. At a temperature of 20 °C, the proposed method for estimating SOC has a mean absolute error (MAE) of 0.72 % and a root mean square error (RMSR) of 0.71 %. In addition, the operational characteristics of the proposed EKBF were validated by simulating its operation at temperatures ranging from 0 °C to 40 °C. Under all conditions, the proposed method maintained an average error rate of 0.40%. The output of the proposed topology was validated using OPAL-RT (OP5700) hardware-in-the-loop real-time simulator. Consequently, the proposed method is preferable for accurate and dependable SOC estimation in real-time battery management system (BMS) applications.

Control & synchronization of a unified chaotic system using an adaptive controller with an extended Kalman–Bucy-filter based Auto-Tuner

A current challenge with adaptive controllers is to define efficient tuning methods of the controller parameters. Unlike linear systems, nonlinear systems may need parameters that are continuously tuned at different operating points to provide stability and desired behaviours. This study aims to develop a solution for tuning proportional–integral–derivative (PID) controller parameters as opposed to changing the operating points of a nonlinear system. Most tuning methods calculate parameters according to the system’s step or frequency response. However, adaptive controllers have self-tuneable parameters or control rules. The proposed algorithm in this paper contains a controller, an estimator, and a reference model, and uses the system model. Unlike the model reference adaptive control method, the proposed controller has tuneable controller parameters estimated by the extended Kalman–Bucy filter. The filter estimates the controller parameters to make the system perform like the auxiliary ideal reference model to ensure minimum-time consumption. Hence, this study aims to develop an algorithm that will automatically calculate controller parameters for each operating point of the controlled chaotic or nonlinear system to minimize settling time at each operating point. The proposed algorithm is implemented in a unified chaotic system in which the estimator and controller of the system run together. Simulation results confirm the performance of the proposed algorithm. In addition, the simulation results provide strong evidence that the proposed algorithm can be an effective tool for controlling nonlinear or chaotic systems.

Speed sensorless Adaptive Power Control for photovoltaic-fed water pump using Extended Kalman–Bucy filter

In this study, a speed sensorless control method is proposed for a photovoltaic water-pump system operating under variable atmospheric conditions. The developed system not only continuously determines the maximum power point of the photovoltaic system but also generates a reference speed appropriate for each power value produced. The MATLAB/Simulink model of the driver system incorporates Adaptive Power Control, Direct Torque Control, a 3-phase induction motor, and the power plant. The determination of the maximum power point of the photovoltaic system and the input reference speed information for the Direct Torque Control under varying atmospheric conditions is accomplished through the application of the Adaptive Power Control algorithm. Notably, the control process operates as a closed-loop system without the requirement of a speed sensor. Instead, the Extended Kalman–Bucy estimator is employed for speed estimation. Empirical evaluation of the proposed method was conducted under constant temperature conditions and varying irradiance values. The results clearly illustrate the consistent operation of the motor, with a constant torque production aligned with each identified maximum power point corresponding to distinct irradiance levels. The control system demonstrates robust performance and operational efficiency across the entire range of operating points. The efficacy of the Kalman estimator in monitoring the reference speed, the utilization of a single solar radiation sensor, and the successful implementation of sensorless operation are significant advantages of the developed adaptive approach. The demonstrated performance of the proposed system showcases its potential for enhancing the efficient and reliable control of photovoltaic water-pump systems in real-world applications.

Mean-square convergent continuous state estimation of randomly switched linear systems with unobservable subsystems and stochastic output noises

This paper studies mean-square (MS) convergent observers for estimating continuous states of randomly switched linear systems (RSLSs) with unobservable subsystems that are subject to stochastic output observation noises. When subsystems are unobservable and switching sequences are random, the classical Kalman–Bucy filters that are applied to observable sub-states are shown to be potentially divergent. It is also shown that unless the switching interval T can be selected to be sufficiently small from the outset, MS convergence may never be achieved, regardless of how the observers for the subsystems are designed. The critical threshold Tmax on T is derived for MS convergent observers to be achievable. Under the condition T<Tmax, this paper introduces design algorithms for subsystem observers to generate a globally MS convergent observer for the entire continuous state. A fundamental design tradeoff between convergence speeds and steady-state estimation errors is analyzed. This paper extends our recent new framework and algorithms for strong convergent observer design in RSLSs by including observation noises, considering multi-output systems, establishing new algorithms for MS convergence, and developing design tradeoff analysis. Examples and a practical case study are presented to illustrate the design procedures, main convergence properties, and error analysis.

PECLOS path-following control of underactuated AUV with multiple disturbances and input constraints

This paper presents a new position error-constrained line-of-sight (PECLOS) path-following (PF) control strategy for an underactuated AUV under multiple disturbances and input constraints. Firstly, considering that many previous studies ignored the influence of stochastic noises, which may degrade the designed controller performance or even lead to instability in practical applications, the multiple disturbances are divided into two categories based on their features, i.e., the slowly time-varying deterministic disturbances and the stochastic noises. On this basis, the time-varying deterministic disturbances are further treated as a single equivalent disturbance and a linear stochastic differential model of underactuated AUV is established by augmenting the equivalent disturbance as an extended state. Secondly, a novel PF control strategy is proposed based on an extended state based Kalman-Bucy filter (ESKBF) and auxiliary dynamic compensators. The PECLOS guidance law is established to maintain the PF errors within the specified time-varying bounds, the ESKBF technology is utilized to estimate and compensate for the equivalent disturbances under the stochastic noises, and the auxiliary dynamic compensators are adopted to overcome the input constraints. Eventually, the stability of the overall closed-loop system is verified. Simulation and comparative analysis are presented to demonstrate the enhanced performance of the presented controller.

Operational Absolutely Optimal Dynamic Control of the Stochastic Differential Plant’s State by Its Output

The problem of synthesizing the average-optimal control law for a dynamic plant subject to random disturbances, if its state variables are measured partially or with random errors, is considered. Using the method of a posteriori sufficient coordinates (SCs), the complexity of constructing the well-known interval-optimal Mortensen controller is described and a much simpler algorithm for finding its operational-optimal analog is obtained. The new controller does not require the solution of the corresponding Bellman equation in inverse time, since it is optimal in the sense of a time-varying criterion. This makes it possible to disregard information about the future behavior of the object and reduces the procedure for finding the dependence of a control on sufficient coordinates to direct-time integration of the Fokker–Planck–Kolmogorov equation and to solving a problem of parametric nonlinear programming. The application of the obtained algorithm is demonstrated by the example of a linear-quadratic-Gaussian problem, as a result of which a new operational version of the well-known separation theorem is formulated. It represents a stochastic control device as a combination of a linear Kalman–Bucy filter and a linear operational-optimal positional controller. The latter differs from the traditional interval-optimal controller by the well-known gain and does not require the solution of the corresponding matrix Riccati equation in inverse time

&lt;p&gt;Estimation of Stochastic Trend on Ukrainian Stock Market&lt;/p&gt;

The paper presents a new approach for price dynamic assessment on the Ukrainian stock market based on a qualitative modification of the Black-Scholes-Merton model for estimating unknown parameters with a stochastic trend. The mean-reverting Ornstein-Uhlenbeck process was chosen as the time-dependent trend parameter since it is both mathematically convenient and has a natural economic interpretation. In order to estimate the parameters of the corresponding stochastic diffusion, the trajectory-fitting technique based on the Kalman-Bucy filter was used. The proposed approach was then tested on UX-index assets.

Log-normalization constant estimation using the ensemble Kalman–Bucy filter with application to high-dimensional models

AbstractIn this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n$ -errors and $\mathbb{L}_n$ -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as $\mathsf{C}(n)\left[t^{1/2}/N^{1/2} + t/N\right]$ or $\mathsf{C}(n)/N^{1/2}$ ( $\mathbb{L}_n$ -errors) and $\mathsf{C}(n)\left[t+t^{1/2}\right]/N$ or $\mathsf{C}(n)/N$ ( $\mathbb{L}_n$ -conditional bias), where t is the time horizon, N is the ensemble size, and $\mathsf{C}(n)$ is a constant that depends only on n, not on N or t. Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.

Localization in Structured Environments with UWB Devices without Acceleration Measurements, and Velocity Estimation Using a Kalman-Bucy Filter.

In this work, a novel scheme for velocity and position estimation in a UWB range-based localization system is proposed. The suggested estimation strategy allows to overcome two main problems typically encountered in the localization systems. The first one is that it can be suitable for use in environments where the GPS signal is not present or where it might fail. The second one is that no accelerometer measurements are needed for the localization task. Moreover, to deal with the velocity estimation problem, a suitable Kalman–Bucy filter is designed and it is compared, experimentally, with a particle filter by showing the features of the two algorithms in order to be used in a localization context. Additionally, further experimental tests are carried out on a suitable developed test setup in order to confirm the goodness of the proposed approach.

Kalman–Bucy filter-based tracking controller design and experimental validations for a quadcopter with parametric uncertainties and disturbances

This paper addresses the problem of trajectory tracking for a quadcopter subject to parametric uncertainties and disturbances. Using the backstepping technique and ignoring parametric uncertainties and disturbances, a nominal controller is proposed for the quadcopter. In order to ensure that the controller is adaptive to parametric uncertainties (i.e. unknown but constant mass and tensor of inertia) and external disturbances, an exponentially stable estimator is designed based on the Kalman–Bucy filter. The closed-loop system resulting from the interconnection of the plant, the controller, and the estimator is such that the backstepping error is uniformly ultimately bounded. Furthermore, simulation and experimental results are presented, validating the stability and robustness properties of the proposed control strategy.

Multilevel Ensemble Kalman–Bucy Filters

In this article we consider the linear filtering problem in continuous-time. We develop and apply multilevel Monte Carlo (MLMC) strategies for ensemble Kalman-Bucy filters (EnKBFs). These filters can be viewed as approximations of conditional McKean-Vlasov-type diffusion processes. They are also interpreted as the continuous-time analogue of the \textit{ensemble Kalman filter}, which has proven to be successful due to its applicability and computational cost. We prove that an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of $\mathcal{O}(\epsilon^2), \ \epsilon>0$ with a cost of order $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of $\mathcal{O}(\epsilon^{-3})$ to achieve an MSE of $\mathcal{O}(\epsilon^2)$. We test our theory on a linear problem, which we motivate through high-dimensional examples of order $\sim \mathcal{O}(10^4)$ and $\mathcal{O}(10^5)$.