Continuous-discrete Extended Kalman Filter Research Articles

Log-Euclidean metric based covariance propagation on SPD manifold for continuous–discrete extended Kalman filtering

For nonlinear systems with continuous dynamic and discrete measurements, a Log-Euclidean metric (LEM) based novel scheme is proposed to refine the covariance integration steps of continuous–discrete Extended Kalman filter (CDEKF). In CDEKF, the covariance differential equation is usually integrated with regular Euclidean matrix operations, which actually ignores the Riemannian structure of underlying space and poses a limit on the further improvement of estimation accuracy. To overcome this drawback, this work proposes to define the covariance variable on the manifold of symmetric positive definite (SPD) matrices and propagate it using the Log-Euclidean metric. To embed the LEM based novel propagation scheme, the manifold integration of the covariance for LEMCDEKF is proposed together with the details of efficient realization, which can integrate the covariance on SPD manifold and avoid the drawback of Euclidean scheme. Numerical simulations certify the new method’s superior accuracy than conventional methods.

Online state estimation of Li-ion batteries using continuous-discrete nonlinear Kalman filters based on a nonlinear simplified electrochemical model

Online monitoring of Lithium-ion batteries (LIBs) during the charge and discharge periods is imperative to characterize their performance, assess their safety, and mitigate their degradation. One way to achieve this goal is to resort to model-based state estimations. Electrochemical based modelling has demonstrated significant potential to predict both the microscopic and the macroscopic battery dynamics. Existing electrochemical models are however continuous-time systems while measurements are performed at discrete time steps. Yet, most estimators are designed based on a fully discretized model. In this study, various types of continuous-discrete nonlinear Kalman filters are developed not only to address this challenge but also to compare their performance in utilizing highly nonlinear and meaningful physical variables. The filters rest on a simplified electrochemical model that accounts for the nonlinearities that stem from the dependence of the parameters on the electrolyte concentration. The performance of the estimators in adaptive and nonadaptive modes is tested for two different driving schedule loads (FUDS and Synthetic random higher C-rate). The results reveal that the estimators can track the real states' values if the filter model matches the real battery, even in the presence of 30 % error in the states' initial conditions. Furthermore, unlike the continuous-discrete extended Kalman filter (CD-EKF), the suggested continuous-discrete unscented Kalman filter (CD-UKF) effectively predicts the actual state values, even in instances where a significant structural mismatch exists between the embedded model and the synthetically generated measurement signal.

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter.

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.

Data-driven parameter estimation in stochastic dynamic neural fields by state-space approach and continuous-discrete extended Kalman filtering

This work examines the state and parameter estimation problems for Stochastic Dynamic Neural Field (SDNF) models. Unlike other studies, we suggest a general approach based on the Galerkin method that reduces the SDNF model to a standard state-space representation. This naturally yields nonlinear continuous-discrete stochastic systems with complicated nonlinear dynamics and linear measurements of a high dimension because of a spatial approximation. This in turn paves a systematic way for a strong integration of nonlinear Bayesian filtering methods into mathematical neuroscience. Because of a high resolution utilized to keep an accurate spatial approximation, a key aspect of nonlinear filters to be derived for the SDNF models is their computational budgets. In this paper, we derive the sequential Itô-Taylor-based continuous-discrete Extended Kalman filter for state and parameter estimation of the SDNF models. The novel method processes each measurement vector in a component-like manner and, hence, avoids an inversion of a large matrix. Thus, the newly-derived sequential estimator is faster and, additionally, more numerically stable compared to the previously developed SDNF-oriented estimators. Besides, the novel filter is an adaptive method, i.e. it solves both problems, which are the state and parameter estimation, in parallel. Finally, the proposed method is applied to numerical examples to provide the comparative study and to show the effectiveness of our result.

Fitting Bayesian Stochastic Differential Equation Models with Mixed Effects through a Filtering Approach

Recent advances in technology contribute to a fast-growing number of studies utilizing intensive longitudinal data, and call for more flexible methods to address the demands that come with them. One issue that arises from collecting longitudinal data from multiple units in time is nested data, where the variability observed in such data is a mixture of within-unit changes and between-unit differences. This article aims to provide a model-fitting approach that simultaneously models the within-unit changes with differential equation models and accounts for between-unit differences with mixed effects. This approach combines a variant of the Kalman filter, the continuous-discrete extended Kalman filter (CDEKF), and the Markov chain Monte Carlo method often employed in the Bayesian framework through the platform Stan. At the same time, it utilizes Stan’s functionality of numerical solvers for the implementation of CDEKF. For an empirical illustration, we applied this method in the context of differential equation models to an empirical dataset to explore the physiological dynamics and co-regulation between couples.

Correlation in Dose-Response to Rapid- and Long-Acting Insulin for People with Type 1 Diabetes.

In diabetes, it can become necessary to switch between pump- and pen-based insulin treatment. This switch involves a translation between rapid- and long-acting insulin analogues. In standard-of-care translation algorithms, a unit-to-unit conversion is applied. However, this simplification may not fit all individuals. In this paper, we investigate the correlation between dose-response to rapid- and long-acting insulin in the same individual, and compare the correlation across individuals. As a measure of dose-response, we estimate the insulin sensitivity in clinical data from 25 subjects with type 1 diabetes. For parameter estimation, we use maximum likelihood with a continuous-discrete extended Kalman filter and Bergman's minimal model. The results show a weak correlation between insulin sensitivity to rapid- and long-acting insulin across individuals. On this sparse data set, the analysis suggests that the standardized unit-to-unit translation between insulin analogues may not benefit all subjects.

Real-Time Estimation of PEMFC Parameters Using a Continuous-Discrete Extended Kalman Filter Derived from a Pseudo Two-Dimensional Model

Proton Exchange Membrane Fuel Cells (PEMFCs) are clean energy conversion devices that are widely used in various energy applications. In most applications, the main challenge is accurately estimating the state of health (SoH) of the PEMFCs during dynamic operating conditions. Moreover, their behavior is affected by numerous physical phenomena such as heat and membrane flooding. This paper proposes the design of an observer to estimate the PEMFC parameters. A state-space model is first built from 2D physical equations solved by a finite difference in a discretized space domain. The discretized dynamic model is then used to design an observer based on the continuous-discrete extended Kalman filter. The observer has been validated experimentally and is used to estimate the parameters of a PEMFC under dynamic operating conditions. For several load variations, the results obtained using the proposed observer accurately characterize the dynamic responses of PEMFC in real-time.

Robust impulsive observer design for infinite‐dimensional cell population balance models

AbstractThe observer design problem for a class of cell population balance models, describing the time evolution of the cell mass density distribution function and the substrate concentration in a continuous‐stirred tank bioreactor with irregular discrete‐time measurements of the cell mass distribution is considered. The model consists of a partial integro‐differential equation coupled with an ordinary differential equation. Using the theory of impulsive systems sufficient conditions for the input‐to‐state‐stability (ISS) of the observation error in the state‐space with respect to the measurement uncertainty are derived in terms of the maximum time between successive measurements and the ISS gain. In absence of measurement uncertainty the convergence conditions imply exponential stability of the observation error dynamics. Besides these rigorous conditions, application‐oriented tuning guidelines are established. The theoretical results are illustrated with numerical simulations including a comparison with a continuous–discrete extended Kalman filter based on the numerical approximation, showing that a similar accuracy is achieved when using a finite‐dimensional approximation of the proposed impulsive observer scheme with a considerably lower computational effort.

Physics Informed Stochastic Grey-Box Model of the Flow-Front in a Vacuum Assisted Resin Transfer Moulding Process with Missing Data

Abstract Real-time fault monitoring and control of the Vacuum Assisted Resin Transfer Moulding production process requires knowledge of the position of the epoxy flow-front inside the mould. Therefore, a fast and accurate flow-front tracking system is highly prized. Physics-informed grey-box models deliver a good trade-off between high fidelity and data-driven black-box models for designing such a flow-front tracking system. In this paper, we propose stochastic differential equations (SDEs) based grey-box model of the flow-front dynamics in the case of missing sensor information. The proposed method uses the finite difference approximation of the spatial domain of the flow-front for estimating the spatial flow pattern of the epoxy. To accommodate for the missing sensor data, we utilize a modified version of the continuous-discrete extended Kalman filter based estimation framework for SDEs that takes into consideration the effective dimension of the measurement space during the identification process. The performance of the method is evaluated for common fault scenarios.

The Supervision of Dough Fermentation Using Image Analysis Complemented by a Continuous Discrete Extended Kalman Filter

Dough fermentation is an important step during the preparation of fermented baking goods. For the supervision of dough fermentation, a continuous-discrete extended Kalman filter was applied, which uses an image analysis system as the measurement. By estimation a fixed number of gas bubbles inside the dough, the radius of an average bubble was determined. A mathematical dough model was used by the extended Kalman filter to estimate the radius of the average bubble, the CO2 concentration of the non-gas dough phase and the number of CO2 molecules in the average bubble. During a fermentation of 50 min, the extended Kalman filter estimated that the average radius increased from 50 µm to 127 µm, the CO2 concentration in the non-gas dough increased to 23 mol/m³, and the CO2 amount in the bubble increased from 0.1 × 10−10 to 4 × 10−10 mol. Also, the specific CO2 production rate was estimated to be in the range from 1.5 × 10−3 to more than 4 × 10−3 mol·m³/kg/s. The advantages of an extended Kalman filter for the supervision of the dough fermentation process are discussed.

Transforming data to information: A parallel hybrid model for real-time state estimation in lignocellulosic ethanol fermentation.

Operating lignocellulosic fermentation processes to produce fuels and chemicals is challenging due to the inherent complexity and variability of the fermentation media. Real‐time monitoring is necessary to compensate for these challenges, but the traditional process monitoring methods fail to deliver actionable information that can be used to implement advanced control strategies. In this study, a hybrid‐modeling approach is presented to monitor cellulose‐to‐ethanol (EtOH) fermentations in real‐time. The hybrid approach uses a continuous‐discrete extended Kalman filter to reconciliate the predictions of a data‐driven model and a kinetic model and to estimate the concentration of glucose (Glu), xylose (Xyl), and EtOH. The data‐driven model is based on partial least squares (PLS) regression and predicts in real‐time the concentration of Glu, Xyl, and EtOH from spectra collected with attenuated total reflectance mid‐infrared spectroscopy. The estimations made by the hybrid approach, the data‐driven models and the internal model were compared in two validation experiments showing that the hybrid model significantly outperformed the PLS and improved the predictions of the internal model. Furthermore, the hybrid model delivered consistent estimates even when disturbances in the measurements occurred, demonstrating the robustness of the method. The consistency of the proposed hybrid model opens the doors towards the implementation of advanced feedback control schemes.

Nonlinear Model Predictive Control for an Exothermic Reaction in an Adiabatic CSTR

We model and simulate an exothermic reaction conducted in an adiabatic continuous stirred tank reactor (CSTR). The system has multiple steady states in part of its operating window. We demonstrate that the three-state model representing the mass and energy balances of the system can be well approximated with a two-state as well as a one-state model. The reduced-order models are relevant for efficient implementation of nonlinear model predictive control (NMPC). We extend the drift term in the system of deterministic ordinary differential equations (ODEs) with a diffusion term such that the process is modeled by a system of stochastic differential equations (SDEs). The SDE representation is well-suited for development of the NMPC that is based on a continuous-discrete extended Kalman filter (CD-EKF) and an optimal control problem. The optimal control problem (OCP) is solved using a multiple-shooting algorithm as the NMPC must be able to stabilize the system at stable as well as unstable steady states. Simulations demonstrate that the NMPC based on a one-state reduced SDE can track any reactor temperature setpoint at stable as well as unstable steady states.

State Estimation in Type 2 Diabetes Using The Continuous-Discrete Unscented Kalman Filter

Using a nonlinear model for the glucose-insulin dynamics in type 2 diabetes, formulated in continuous-time as a stochastic differential equation, we seek to estimate the system states and parameters based only on discrete-time self-monitored blood glucose measurements of fasting glucose and the known exogenous insulin dose. This is done by means of continuous-discrete unscented Kalman filtering. The results are compared to an implementation of a continuous-discrete extended Kalman filter. Simulations show that it is possible to estimate all states with good accuracy using the CD-UKF, while it is also possible to estimate one unknown parameter at the same time. Further simulations show that increasing the sample rate makes it possible to estimate more parameters, given that the meal intake of the patient is known perfectly.

A Comparative Study of Kalman-like Filters for State Estimation of Turning Aircraft in Presence of Glint Noise

This paper continues the study started by Kulikov and Kulikova on state estimation accuracies of various Kalman-like filtering techniques in target tracking scenarios with non-Gaussian noise in 2018. The cited authors examined a number of methods, which are grounded in the minimum-variance or maximum-correntropy criteria and cover extended-, cubature- and unscented-type Kalman filters, in the well-known turning aircraft scenario with impulsive (shot) noise or mixed-Gaussian one. Despite the success of the maximum-correntropy-based filtering methods reported on estimation of linear discrete-time stochastic systems in literature, those case studies expose the superiority of the cubature and unscented Kalman filters towards various extended Kalman methods designed in the minimum-variance sense or grounded in the maximum-correntropy criterion within the mentioned target tracking scenarios. Here, we extend that examination to the turning aircraft scenario with glint noise, which is simulated by a sum of two zero-mean Gaussian variables with difference covariances. In particular, our study reveals a valued potential of the maximum-correntropy-based accurate continuous-discrete extended Kalman filters devised by the above authors in this glint noise state estimation environment.

Square-root accurate continuous-discrete extended-unscented Kalman filtering methods with embedded orthogonal and J-orthogonal QR decompositions for estimation of nonlinear continuous-time stochastic models in radar tracking

This paper presents a number of new state estimation algorithms, which unify the best features of the accurate continuous-discrete extended and unscented Kalman filters in treating nonlinear continuous-time stochastic systems with discrete measurements. In particular, our mixed-type algorithms succeed in estimating continuous-discrete stochastic systems with nonlinear and/or nondifferentiable measurements. The main weakness of these methods is the need for the Cholesky decomposition of predicted covariance matrices. Such a factorization is highly sensitive to numerical integration and round-off errors committed, which may result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually solved in the form of square-root filtering implementations, which propagate not the covariance matrix but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of our mixed-type methods to high-dimensional stochastic systems preclude from designing conventional square-root filters. We address the mentioned issue with one-rank Cholesky factor updates or with hyperbolic QR transforms used for yielding J-orthogonal square-root filters. These novel algorithms are justified theoretically and examined and compared numerically to the non-square-root one in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, in the presence of Gaussian or glint noise.

Nonlinear Model Predictive Control for Disturbance Rejection in Isoenergetic-isochoric Flash Processes

Abstract We present a nonlinear model predictive control (NMPC) algorithm for semi-explicit index-1 stochastic differential-algebraic equations. It is natural to model isoenergetic-isochoric (constant energy-constant volume) flash processes with such equations. The algorithm uses the continuous-discrete extended Kalman filter (EKF) for state estimation, and it uses a single-shooting method to solve the involved optimal control problems. It computes the gradients with an adjoint method. The isoenergetic-isochoric flash is also called the UV flash, and it is important to rigorous models of phase equilibrium processes because it is a mathematical statement of the second law of thermodynamics. NMPC algorithms for UV flash processes are therefore relevant to both safe and economical operation of phase equilibrium processes such as flash separation, distillation, two-phase flow in pipes, and oil production. We design the NMPC algorithm for disturbance rejection, and we therefore augment the state vector with the unknown disturbance variables in the continuous-discrete EKF. We present a numerical example of economical NMPC of a UV flash separation process. It involves output constraints and the estimation of an unknown and unmeasured disturbance. The computation time of the NMPC algorithm does not exceed 3.14 s in any of the 5 min control intervals. This indicates that real-time NMPC of UV flash separation processes is computationally feasible.

The p53–Mdm2 regulation relationship under different radiation doses based on the continuous–discrete extended Kalman filter algorithm

The tumour suppressor gene p53 plays a key role in cell response to DNA damage, and the p53–Mdm2 regulation relationship is crucial for the expression of p53. In this paper, based on gene expression time-series data of human leukaemia cells after exposure to ionising radiation, with ionising radiation as input, a nonlinear continuous time-delay dynamic stochastic mathematical model of a p53–Mdm2 network is established using a continuous–discrete extended Kalman filter algorithm. The accuracy of the established model is then validated. Numerical simulation is used to simulate the dynamic regulation in p53–Mdm2 networks for low, medium, and high ionising radiation doses. The results show that the proposed algorithm is convergent and that the error rate of the model is only 1.19%. In addition, the model can simulate accurately the response of the p53–Mdm2 network to different doses of ionising radiation. The methods proposed in this paper can supply the foundation for research on the dynamics of the p53–Mdm2 gene regulation relationship and play a guiding role in research on the p53–Mdm2 response process after DNA damage under different doses of ionising radiation.

Accurate State Estimation in Continuous-Discrete Stochastic State-Space Systems With Nonlinear or Nondifferentiable Observations

This paper presents a novel method of nonlinear Kalman filtering, which unifies the best features of the accurate continuous-discrete extended and cubature Kalman filters. More precisely, the time updates in the discussed state estimator are done as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All this allows accurate predictions of the state mean and covariance matrix to be combined with accurate measurement updates. Moreover, the new filter is particularly effective for continuous-discrete stochastic systems with nonlinear and/or nondifferentiable observations. The efficiency of this mixed-type method is shown in comparison to the performance of the original accurate continuous-discrete extended and cubature Kalman filters in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, with sufficiently long sampling times.

Estimating Lithium-Ion Battery State of Charge and Parameters Using a Continuous-Discrete Extended Kalman Filter

A real-time determination of battery parameters is challenging because batteries are non-linear, time-varying systems. The transient behaviour of lithium-ion batteries is modelled by a Thevenin-equivalent circuit with two time constants characterising activation and concentration polarization. An experimental approach is proposed for directly determining battery parameters as a function of physical quantities. The model’s parameters are a function of the state of charge and of the discharge rate. These can be expressed by regression equations in the model to derive a continuous-discrete extended Kalman estimator of the state of charge and of other parameters. This technique is based on numerical integration of the ordinary differential equations to predict the state of the stochastic dynamic system and the corresponding error covariance matrix. Then a standard correction step of the extended Kalman filter (EKF) is applied to increase the accuracy of estimated parameters. Simulations resulting from this proposed estimator model were compared with experimental results under a variety of operating scenarios—analysis of the results demonstrate the accuracy of the estimator for correctly identifying battery parameters.

Do the Cubature and Unscented Kalman Filtering Methods Outperform Always the Extended Kalman Filter ?

This paper elaborates three well-known state estimators, which are used extensively in practice. These are the classical continuous-discrete extended Kalman filter (EKF) and the continuous-discrete cubature Kalman filtering (CKF) and unscented Kalman filtering (UKF) algorithms designed recently. Nowadays, it is commonly accepted that the contemporary filters always outperform the traditional EKF in the accuracy of state estimation because of their higher-order approximation of the mean of propagated Gaussian density in the time- and measurement-update steps of the modern techniques. However, the present paper specifies this commonly accepted opinion and shows that despite the mentioned theoretical fact the EKF may outperform the CKF and UKF methods in the accuracy of state estimation when the stochastic system under consideration exposes a stiff behavior. That is why stiff stochastic models are difficult to deal with and require effective state estimation algorithms to be devised yet.