Kalman-Bucy Filter Research Articles

Filtering with degenerate observation noise: A stochastic approximation approach

This paper is concerned with a general filtering scheme of a continuous-time dynamic system in which the state is not completely observable. The observation process consists of a function of the state with additive noise. Typically, such noise is assumed to be non-degenerate. In this case, various filtering schemes can be developed. For example, in a linear case, the Kalman–Bucy (KB) filter applies and leads to a recursive filtering equation for the conditional expectation of the state given the observation up to time t. Nevertheless, in applications, only some state variables are directly observable and the rest are not. This gives rise to filtering with degenerate observation noise. In this case, traditional filtering schemes fail. This paper develops a viable scheme to address possible degenerate observation noise. In this paper, we propose a recursive filtering equation in which the gain matrix is a matrix-valued parameter to be determined. We adopt the Monte Carlo training procedure used for deep filtering to determine the best gain matrix. In particular, given the state and observation equations, we generate their Monte Carlo samples. Given a gain matrix, we generate the corresponding state estimation samples, which leads to the error function of the state and its estimation. The problem is to choose the gain matrix to minimize the error function. We develop a stochastic approximation method for such a minimization task; we term the procedure the SA filter, where SA stands for stochastic approximation. We focus on computational experiments and demonstrate the performance of the SA filter and its robustness. We also compare the SA filter with the (extended) Kalman–Bucy filter and the deep filter in both linear and nonlinear models.

Univariate and Multivariate Time Series Modeling using a Harmonic Decomposition Methodology

This paper contributes by developing an univariate and multivariate harmonic decomposition methodology to model time series.The models are stated in a state space representation derived from a Fourier series analysis to describe an arbitrary signal. The frequency values of the harmonic content are used to define state variables in order to describe a signal through a time-varying linear state space model, which serves to synthesize an optimal state observer (Kalman-Bucy filter). Once the observer converges and the states (harmonics) become constant, the observer model can be used to predict the signal, i.e., a time series forecasting can be performed.The preocedure can be developed for the univariate or multivariate case of time series modeling, where in the last one, a statistical analysis is used to determine which variables should be taken into account to obtain a more accurate model. The proposed modeling approach is successfully applied for the modeling and forecast of the time series of electrical power demand and a wind/solar profile.

Optimal guidance and estimation of a 2D diffusion–advection process by a team of mobile sensors

This paper describes an optimization framework to design guidance for a possibly heterogeneous team of multiple mobile sensors to estimate a spatiotemporal process modeled by a 2D diffusion–advection process. Owing to the abstract linear system representation of the process, we apply the Kalman–Bucy filter for estimation, where the sensors provide linear outputs. We propose an optimization problem that minimizes the sum of the trace of the covariance operator of the Kalman–Bucy filter and a generic mobility cost of the mobile sensors, subject to the sensors’ motion modeled by linear dynamics. We establish the existence of a solution to this problem. Moreover, we prove convergence to the exact optimal solution of the approximate optimal solution. That is, when evaluating these two solutions using the original cost function, the difference becomes arbitrarily small as the approximation gets finer. To compute the approximate solution, we use Pontryagin’s minimum principle after approximating the infinite-dimensional terms originating from the diffusion–advection process. The approximate solution is applied in simulation to analyze how a single mobile sensor’s performance depends on two important parameters: sensor noise variance and mobility penalty. We also illustrate the application of the framework to multiple sensors, in particular the performance of a heterogeneous team of sensors.

Derivation of ensemble Kalman–Bucy filters with unbounded nonlinear coefficients

We provide a rigorous derivation of the ensemble Kalman–Bucy filter as well as the ensemble transform Kalman–Bucy filter in case of nonlinear, unbounded model and observation operators. We identify them as the continuous time limit of the discrete-time ensemble Kalman filter and the ensemble square root filters, respectively, together with concrete convergence rates in terms of the discretisation step size. Simultaneously, we establish well-posedness as well as accuracy of both the continuous-time and the discrete-time filtering algorithms.

Numerical and experimental study of an active control logic for modifying the acoustic performance of single-layer panels

This work proposes an active control logic that uses a linear quadratic regulator (LQR) controller and a Kalman-Bucy (KB) state observer to improve the acoustic performance of single-layer panels. When the panel is subjected to an acoustic excitation, the proposed strategy can automatically adapt to the acoustic disturbance by tuning some of the decisive weighting factors in the LQR and the KB filter according to the spectrum of the signals from sensors. This control acts on the panel with PZT patches as actuators and accelerometers as sensors, forming a smart structure. The vibroacoustic model of the smart panel is formulated using modal functions, from which the transmitted power and the transmission loss of the panel are derived. Accordingly, the control strategy is designed using the state-space representation under modal coordinates, during which the spillover effect is considered and the placement of actuators and sensors is optimized by a modified modal H2 norm approach. It is noticeable that several advanced techniques in the active control of noise and vibration are implemented in this development of the smart panel. Following this, numerical and experimental studies oriented to the effectiveness of the proposed control logic are performed. While the numerical simulations are carried out under the assumption that the two sides of the panel are a reverberant and a free field, respectively, the experiments are conducted using the test equipment called Noise-Box, whose inner space is considered able to simulate a field incidence. Two scenarios are provided: one for the monotone at 500 Hz or 1,000 Hz; the other for the white noise within 750 Hz – 1,000 Hz. The results validate that the active control improves the acoustic performance of the panel for sound insulation, whether the acoustic disturbance is a monotone of any frequency or a band-limited noise with a given frequency range.

IN-SAMPLE ASYMPTOTICS AND ACROSS-SAMPLE EFFICIENCY GAINS FOR HIGH FREQUENCY DATA STATISTICS

We revisit in-sample asymptotic analysis extensively used in the realized volatility literature. We show that there are gains to be made in estimating current realized volatility from considering realizations in prior periods. The weighting schemes also relate to Kalman-Bucy filters, although our approach is non-Gaussian and model-free. We derive theoretical results for a broad class of processes pertaining to volatility, higher moments, and leverage. The paper also contains a Monte Carlo simulation study showing the benefits of across-sample combinations.

Multiple model adaptive postprandial glucose control of type 1 diabetes

In this work, the adaptive regulation of blood glucose (BG) in type I diabetic (T1D) patients is considered by developing aMultiple Model Adaptive Control (MMAC), where its estimation is based on Magdelaine'slong-term glucose-insulin Model.The (MMAC) is built using a bank of Kalman- Bucy Filters (KBFs)with optimal state feedback controllers. Each KBF is based on a particular value of the equilibrium point for which, the optimal Linear Quadratic Servo (LQ-Servo) controller is designed. The total state estimation is resolved by the probabilistic weighted sum of the produced outputs of all filters based on measured glucose signal. Simulation results show that MMAC is capable of providing reliable estimation and regulation of insulin delivery. Moreover, the performance of the controlled glucose/insulin is improved by 99% compared with that when using a single KBF. The MMAC has accurately identified the glucose signal corresponding to the hypothesis models with an average accuracy of 96.4% for 5 tested patients. Robust performance has been tested with different initial conditions and disturbance.

Linear Kalman-Bucy filter with vector autoregressive signal and noise

The linear Kalman-Bucy filter problem for a system, at that a signal and a noise are vector independent stationary autoregressive processes with orders larger than 1, is investigated. The recurrent equations for filter and its error are delivered. The optimal way of the initial data definition is proposed. Some numerical examples are given. In one of them the algorithm leads to a stationary behavior at infinity. In the other example the Kalman- Bucy filter is impossible because the filter error goes to infinity. A behavior of a signal and its error is illustrated by a simulation of a signal and a noise as vector Gaussian stationary autoregressive processes. The simulation supports theoretical conclusions.

Linear quadratic Gaussian Stackelberg game under asymmetric information patterns

This paper investigates the linear quadratic Gaussian Stackelberg game under asymmetric information. Two decision makers implement control strategies using different information patterns: The follower uses its observation information to design its strategy, whereas the leader implements its strategy using global information. For this problem, the separation principle becomes invalid. Instead, we apply the classical variation method to derive conditions to be satisfied by both decision makers. We show that the solution attributes to solving a two-point boundary value problem of stochastic version whose drift terms contain some conditional mean terms. We then propose a layered calculation method to solve this problem. In the inner layer calculation, the adjoint state variable’s estimate is expressed as a linear functional of the state estimate. In the outer layer calculation, the adjoint state variable is expressed as a linear functional of the state and its estimate. We also verify that the Kalman–Bucy filter works for computing the state estimate. The result is also applied to dynamic duopoly with sticky prices and pursuit–evasion problem.

County-Level Social Distancing and Policy Impact in the United States: A Dynamical Systems Model.

BackgroundSocial distancing and public policy have been crucial for minimizing the spread of SARS-CoV-2 in the United States. Publicly available, county-level time series data on mobility are derived from individual devices with global positioning systems, providing a variety of indices of social distancing behavior per day. Such indices allow a fine-grained approach to modeling public behavior during the pandemic. Previous studies of social distancing and policy have not accounted for the occurrence of pre-policy social distancing and other dynamics reflected in the long-term trajectories of public mobility data.ObjectiveWe propose a differential equation state-space model of county-level social distancing that accounts for distancing behavior leading up to the first official policies, equilibrium dynamics reflected in the long-term trajectories of mobility, and the specific impacts of four kinds of policy. The model is fit to each US county individually, producing a nationwide data set of novel estimated mobility indices.MethodsA differential equation model was fit to three indicators of mobility for each of 3054 counties, with T=100 occasions per county of the following: distance traveled, visitations to key sites, and the log number of interpersonal encounters. The indicators were highly correlated and assumed to share common underlying latent trajectory, dynamics, and responses to policy. Maximum likelihood estimation with the Kalman-Bucy filter was used to estimate the model parameters. Bivariate distributional plots and descriptive statistics were used to examine the resulting county-level parameter estimates. The association of chronology with policy impact was also considered.ResultsMobility dynamics show moderate correlations with two census covariates: population density (Spearman r ranging from 0.11 to 0.31) and median household income (Spearman r ranging from –0.03 to 0.39). Stay-at-home order effects were negatively correlated with both (r=–0.37 and r=–0.38, respectively), while the effects of the ban on all gatherings were positively correlated with both (r=0.51, r=0.39). Chronological ordering of policies was a moderate to strong determinant of their effect per county (Spearman r ranging from –0.12 to –0.56), with earlier policies accounting for most of the change in mobility, and later policies having little or no additional effect.ConclusionsChronological ordering, population density, and median household income were all associated with policy impact. The stay-at-home order and the ban on gatherings had the largest impacts on mobility on average. The model is implemented in a graphical online app for exploring county-level statistics and running counterfactual simulations. Future studies can incorporate the model-derived indices of social distancing and policy impacts as important social determinants of COVID-19 health outcomes.

Application of the Kalman Filter in Functional Magnetic Resonance Image Data

The Kalman-Bucy filter was applied on the preprocessing of the functional magnetic resonance image-fMRI. Numerical simulations of hemodynamic response added Gaussian noise were performed to evaluate the performance of the filter. After the proceeding was applied in auditory real data. The Kohonen’s self-organized map was employed as tools to compare the performance of the Kalman’s filter with another type of pre-processing. The results of the application of Kalman-Bucy filter for simulated data and real auditory data showed that it can be used as a tool in the temporal filtering step in fMRI data.

Resilient Secondary Voltage Control of Islanded Microgrids: An ESKBF-Based Distributed Fast Terminal Sliding Mode Control Approach

This paper proposes a distributed secondary voltage control method based on extended state Kalman-Bucy filter (ESKBF) and fast terminal sliding mode (FTSM) control for the resilient operation of an islanded microgrid (MG) with inverter-based distributed generations (DGs). To tackle the co-existence of multiple uncertainties, a unified modelling framework is proposed to represent the set of different types of disturbances, including parameter perturbation, measurement noise, and immeasurably external variables, by an extended state method. Kalman-Bucy filter is then applied to accurately estimate the state information of the extended DG model. In addition, based on the accurate estimation, a fast terminal sliding mode (FTSM) surface with terminal attractors is designed to maintain the system stability and accelerate the convergence of consensus tracking, which significantly improves the performance of secondary voltage control under both normal and plug-and-play operation. Finally, case studies are conducted in both MATLAB/Simulink and an experimental testbed to demonstrate the effectiveness of the proposed method.

Asymptotic properties of linear filter for deterministic processes

It is known that Kalman–Bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability. In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system, i.e., for deterministic processes. The earlier stability results cannot be applied for this case, as the system is not controllable. We further show that the optimal linear filter for a general class of non-Gaussian initial conditions is asymptotically proximal to Kalman–Bucy filter. It is also shown that the filter corresponding to non-zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of Gaussian initial conditions.

Extending the Theory of Liptser-Shiryaev Filters

Two types of approximate suboptimal Liptser-Shiryaev filters (LSFs) are developed. The first type is based on the generalized Kalman-Bucy filter; the second type, on the parameterization of the a posteriori distribution using the orthogonal expansion method and the quasi-moment method. In principle, the design methods of the LSFs (see below) can be used for obtaining an almost optimal filter with any degree of accuracy. The higher the maximum order of the orthogonal expansion parameters under consideration is, the higher the accuracy of approximation to the optimal estimate will be.

Application of Kalman-Bucy filter for vessel traffic control systems in the northern sea route

The article considers the method of ice density concentration prediction based on the implementation of the Kalman-Bucy filter using geospatial data. The developed algorithm of data filtering allows us to assess the effective use of this approach in predicting the density of ice in the Arctic. The key aspects of the model application for the construction of optimal and safe routes of vessels in the waters of the Northern sea route are described.

RETRACTED ARTICLE: A Sly Salvage of Semantic Web Content with Insistence of Low Precision and Low Recall

Semantic Web content extracting are the augmentation of the present web where the data is given in the better importance and allowing users to work close by close by utilizing the assets of web. Due to expanding of web content at speedier rate, it is difficult to adapt up to new procedures of separating to create the exact data in light of the user query. Later, number of scientists has taking a shot at enhancing the aftereffects of Web content mining by abusing Web Semantic Structure and can be executed in the web expresses through machine-processable data to bolsters the errands of users. But these methods bring about low precision [unseemly data from different inquiry result (equivocal)] and getting of off base data (i.e. low recall) with high surfing time. Therefore, the paper proposes the Semantic Web content mining utilizing two ways to deal with dodge low precision and low recall: the one is Kalman–Bucy filters, to channel the comparable information; the next is adaptive Helmholtz machine, to get the exact data in light of the user query. The proposed Kalman–Bucy filtering approach with adaptive Helmholtz machine execution is be assessed and contrasted with existing methodologies regarding retrieval accuracy and surfing time by dual predicted output and the optimized information retrieval rather than probability.

Extended State Filter Based Disturbance and Uncertainty Mitigation for Nonlinear Uncertain Systems With Application to Fuel Cell Temperature Control

The filter design for nonlinear uncertain systems is quite challenging since efficient estimation is required against stochastic noises, nonlinear uncertain dynamics as well as their concurrent effects. To this end, this article develops a novel filter algorithm by augmenting the disturbance as well as unknown nonlinear dynamics as an extended state and constructing consistent Kalman–Bucy algorithm. The proposed extended state based Kalman–Bucy filter (KBF) is shown to be of bounded estimation error, and the estimation accuracy can be online evaluated. More importantly, the estimation of asymptotic minimum variance is realized in condition that the changing rate of uncertainty approaches to zero. Therefore, the proposed extended state filter enables effective mitigation of disturbance and unknown nonlinear dynamics in real time by feedback control. The proposed algorithm is experimentally verified via a temperature control application in proton exchange membrane fuel cell, in which the thermocouple noise and the electrochemical uncertainty are seriously presented. The temperature variation of the extended state based KBF-based control is greatly reduced, in comparison with the conventional control. The results in this article depict a promising prospect of the proposed method for industrial control applications to handle both noises and nonlinear uncertain dynamics.

Kalman filter and satellite attitude control system analytical design

The article considers a certain form of suboptimal Kalman-Bucy filter (KBF), with bounded grows of memory (FBGM) and its application for satellite control system analytical design. The FBGM is robust and simple. Due to these qualities it can be easily practically implemented. This is especially important for airspace applications, where on-board computers (OBC), despite of the huge progress achieved in their characteristics, still have restricted resources. Since FBGM in the steady state can be made optimal KBF, it is used for, so called, analytical design (AD) of ACS, to estimate its potentially achievable performance in an idealistic case. This case assumes that all restrictions and constraints are removed and not taking into account as for the real system design process. Various schemes of filter usage in conjunction with the controller in closed negative feedback control loop are discussed. The emphasis is on analytical design; however, taking into account engineering aspects, the solution found can be implemented in real system as well.

Posterior contraction rates for non-parametric state and drift estimation

<p style='text-indent:20px;'>We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman–Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upon scale separation and a slow manifold approximation.

Efficient Temperature Profile Estimation for Silicon Wafers based on Subspace Observers

This work proposes a computationally efficient observation algorithm for the surface temperature profile of heated silicon wafers. The observer exploits the fact that only a few modes of the original large-order model are unstable or slowly converging. In this case, it suffices to modify only these modes by the observer’s output feedback gain. Compared to classical observation techniques, the proposed method allows to compute the feedback gain for a state space of lower dimension, which reduces computational complexity. A comparison of the approach with the extended Kalman-Bucy filter using experimental data shows its appealing performance.