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.

A Novel Prediction Method of Transfer-Assisted Action Oriented to Individual Differences for the Excretion Care Robot.

The excretion care robot's (ECR) accurate recognition of transfer-assisted actions is crucial during its usage. However, transfer action recognition is a challenging task, especially since the differentiation of actions seriously affects its recognition speed, robustness, and generalization ability. We propose a novel approach for transfer action recognition assisted by a bidirectional long- and short-term memory (Bi-LSTM) network combined with a multi-head attention mechanism. Firstly, we utilize posture sensors to detect human movements and establish a lightweight three-dimensional (3D) model of the lower limbs. In particular, we adopt a discrete extended Kalman filter (DEKF) to improve the accuracy and foresight of pose solving. Then, we construct an action prediction model that incorporates a fused Bi-LSTM with Multi-head attention (MHA Bi-LSTM). The MHA extracts key information related to differentiated movements from different dimensions and assigns varying weights. Utilizing the Bi-LSTM network effectively combines past and future information to enhance the prediction results of differentiated actions. Finally, comparisons were made by three subjects in the proposed method and with two other time series based neural network models. The reliability of the MHA Bi-LSTM method was verified. These experimental results show that the introduced MHA Bi-LSTM model has a higher accuracy in predicting posture sensor-based excretory care actions. Our method provides a promising approach for handling transfer-assisted action individual differentiation in excretion care tasks.

Linearization-based state-transition model for the discrete extended Kalman filter applied to multibody simulations

This study investigates the discrete extended Kalman filter as applied to multibody systems and focuses on accurate formulation of the state-transition model in the framework. The proposed state-transition model is based on the coordinate-partitioning method and linearization of the multibody equations of motion. The approach utilizes the synergies between the integration of states and estimator covariances without overly simplifying the integrator structure. The proposed method is analyzed with a forward dynamics analysis of a four-bar mechanism. The results show that the stability of the state-transition model in the forward dynamics analysis is significantly enhanced with the proposed method compared with the forward Euler-based methods. The computational efficiency of the novel method was significantly lower in comparison to forward Euler-based methods, which was found to be mainly due to the computation of the Jacobian matrix of the nonlinear state equation. However, the increase in computational cost can be considered acceptable in Kalman-filtering applications, where the exact Jacobian of the state equation is needed.

Comparison of EKF and TSO for Health Monitoring of a Textile-Based Heater Structure and its Control

In this article, the conception, design, and implementation of a discrete extended Kalman filter and a Takagi–Sugeno observer are introduced, and an innovative model description of an embroidered heater structure is presented. The observers are compared based on the algorithm complexity with the indicator of the number of arithmetic operations and the parameter error of the textile-based heater structure for health monitoring through virtual sensing. The proposed schemes allow an estimation of the heater temperature to guarantee safety by preventing overheating without a direct measurement. An estimation strategy is necessary because measurements of the temperature of the whole heater structure are usually not available, but this knowledge is vital for health monitoring purposes or for a safe shutdown if necessary. A cascaded sliding-mode control strategy using an integral surface for the electrical current and the temperature is presented. Measurements validate the utilized model. The effectiveness of the observers is justified by hardware-in-the-loop simulations and measurements.

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.

Estimating the Characteristic Curve of a Directional Control Valve in a Combined Multibody and Hydraulic System Using an Augmented Discrete Extended Kalman Filter.

The estimation of the parameters of a simulation model such that the model’s behaviour matches closely with reality can be a cumbersome task. This is due to the fact that a number of model parameters cannot be directly measured, and such parameters might change during the course of operation in a real system. Friction between different machine components is one example of these parameters. This can be due to a number of reasons, such as wear. Nevertheless, if one is able to accurately define all necessary parameters, essential information about the performance of the system machinery can be acquired. This information can be, in turn, utilised for product-specific tuning or predictive maintenance. To estimate parameters, the augmented discrete extended Kalman filter with a curve fitting method can be used, as demonstrated in this paper. In this study, the proposed estimation algorithm is applied to estimate the characteristic curves of a directional control valve in a four-bar mechanism actuated by a fluid power system. The mechanism is modelled by using the double-step semi-recursive multibody formulation, whereas the fluid power system under study is modelled by employing the lumped fluid theory. In practise, the characteristic curves of a directional control valve is described by three to six data control points of a third-order B-spline curve in the augmented discrete extended Kalman filter. The results demonstrate that the highly non-linear unknown characteristic curves can be estimated by using the proposed parameter estimation algorithm. It is also demonstrated that the root mean square error associated with the estimation of the characteristic curve is 0.08% with respect to the real model. In addition, all the errors in the estimated states and parameters of the system are within the 95% confidence interval. The estimation of the characteristic curve in a hydraulic valve can provide essential information for performance monitoring and maintenance applications.

Estimation of Angular Velocity and Position of a Permanent Magnet Synchronous Motor Using Discrete-Time Extended Kalman Filter and Hybrid Extended Kalman Filter

In this paper we analyze the permanent magnet synchronous motor. The state equations governing the permanent magnet synchronous motor are nonlinear, so the angular position and velocity of its rotor are estimated using nonlinear filters. Since the current passing through the armature is measurable, this measurement is used to estimate system states (i.e. angular position and angular velocity). Discrete extended Kalman filter and hybrid extended Kalman filter algorithms are presented. The comparison between simulation results of rotor’s position and velocity estimation using discrete extended Kalman filter and hybrid Kalman filter shows that hybrid Kalman filter estimator has better performance.

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.

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 continuous–discrete unscented Kalman filtering for estimation of nonlinear continuous-time stochastic models in radar tracking

This paper presents a new state estimation technology grounded in the unscented Kalman filtering for nonlinear continuous-time stochastic systems. The resulting accurate continuous–discrete unscented Kalman filter is based on adaptive solvers with automatic global error control for treating numerically the moment differential equations arising in the mean and covariance calculation of propagated Gaussian density. It is intended for an accurate and robust state estimation in nonlinear continuous–discrete stochastic systems of various sorts, including in radar tracking models. This new filter is examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn. The latter is considered to be a challenging one for testing nonlinear filtering algorithms. For comparison, we also examine such efficient state estimators as the accurate continuous–discrete extended Kalman filter, the continuous–discrete unscented Kalman filter and the mixed-type accurate continuous–discrete extended-unscented Kalman filter designed earlier, but further modified in the present study. The comparison is fulfilled in terms of accuracy and efficiency of estimating the state in the mentioned air traffic control scenario.

The continuous–discrete extended Kalman filter revisited

Abstract This paper elaborates a new approach to nonlinear filtering grounded in an accurate implementation of the continuous–discrete extended Kalman filter for estimating stochastic dynamic systems. It implies that the moment differential equations for calculation of the predicted state mean and error covariance of propagated Gaussian density are solved accurately, i.e., with negligible errors. The latter allows the total error of the extended Kalman filter to be reduced significantly and results in a new accurate continuous–discrete extended Kalman filtering method. In addition, this filter exploits the scaled local and global error controls to avoid any comparison of different physical units. The designed state estimator is compared numerically with continuous–discrete unscented and cubature Kalman filters to expose its practical efficiency. The problem of long waiting times (i.e., infrequent measurements) arisen in chemical and other engineering is also addressed.

Accurate cubature and extended Kalman filtering methods for estimating continuous-time nonlinear stochastic systems with discrete measurements

This paper further advances the idea of accurate Gaussian filtering towards efficient cubature Kalman filters for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the moment differential equations describing evolution of the predicted mean and covariance of the propagated Gaussian density in time are solved accurately, i.e. with negligible error. The latter allows the total error of the cubature Kalman filtering to be reduced significantly and results in a new accurate continuous–discrete cubature Kalman filtering method. At the same time, we revise the earlier developed version of the accurate continuous–discrete extended Kalman filter by amending the involved iteration and relaxing the utilized global error control mechanism. In addition, we build a mixed-type method, which unifies the best features of the accurate continuous–discrete extended and cubature Kalman filters. More precisely, the time updates are done in this state estimator 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 these are examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, and compared to the state-of-the-art cubature Kalman filters.

Online state and input force estimation for multibody models employing extended Kalman filtering

This paper discusses the use of Subsystem Global Modal Parameterization (SS-GMP) reduced multibody models in an augmented discrete extended Kalman filter (A-DEKF) to generate a general formalism for online coupled state/input estimation in mechanisms. The SS-GMP approach is proposed to reduce a general multibody model of a mechanical system into a real-time capable model without considerable loss in accuracy. In order to use these reduced models with an extended Kalman filter, the necessary derivatives of this model are provided. An exponential integration scheme is used to discretize the model in order to be compatible with discrete time filters. Finally, the augmented approach is used for the estimation, in which the unknown external forces are considered as additional states to be estimated. The proposed approach is validated numerically and compared to three other filtering approaches. The validation demonstrates that the proposed approach provides accurate results while still maintaining real-time performance.

Prediction models for wind speed at turbine locations in a wind farm

ABSTRACTIn wind farms, individual turbines disturb the wind field by generating wakes that influence other turbines in the farm. From a control point of view, there is an interest in dynamic optimization of the balance between fatigue and production, and an understanding of the relationship between turbines manifested through the wind field is hence required. This paper develops models for this relationship. The result is based on two new contributions: the first is related to the estimation of effective wind speeds, which serves as a basis for the second contribution to wind speed prediction models.Based on standard turbine measurements such as rotor speed and power produced, an effective wind speed, which represents the wind field averaged over the rotor disc, is derived. The effective wind speed estimator is based on a continuous–discrete extended Kalman filter that takes advantage of nonlinear time varying turbulence models. The estimator includes a nonlinear time varying wind speed model, which compared with literature results in an adaptive filter. Given the estimated effective wind speed, it is possible to establish wind speed prediction models by system identification. As the prediction models are based on the result related to effective wind speed, it is possible to predict wind speeds at neighboring turbines, with a separation of over 700 m, up to 1 min ahead reducing the error by 30% compared with a persistence method. The methodological results are demonstrated on data from an off‐shore wind farm. Copyright © 2011 John Wiley & Sons, Ltd.

Multirate Asynchronous State and Parameter Estimation for Centrifugal Compressors

AbstractA number of strategies for state and parameter estimation for a centrifugal compressor system are investigated using a nonlinear dynamical model. It is found that high frequency measurements can be combined with lower frequency measurements to achieve high fidelity state estimation for stable equilibria, limit cycles, as well as transients using either a multirate discrete extended Kalman filter or a data fusion based algorithm with one independent estimator per measurement. When there is a change in one of the parameters of the system only a filter designed to adapt to such changes produces estimates with acceptable errors for a locally stable equilibrium.

Measuring Time-Varying $I/Q$ Impairments in Digital Transmitters

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Impairments affecting the baseband modulator of digital transmitters, which are commonly called <formula formulatype="inline"><tex Notation="TeX">$I/Q$</tex></formula> impairments, are responsible for the deviations of symbols from their original position on the <formula formulatype="inline"><tex Notation="TeX">$I/Q$</tex></formula> diagram. They cause a reduction in the noise margin and ultimately worsen the probability of error. This is why their measurement is so important in all stages in the life cycle of digital transmitters. This paper presents an original method for the evaluation of the <formula formulatype="inline"><tex Notation="TeX">$I/Q$</tex></formula> impairments based on a discrete extended Kalman filter (DEKF). In the same way as alternative solutions already presented in the literature, the method exhibits good performance in the presence of time-invariant impairments. Differently from them, it allows accurate measurements of time-varying impairments, thus making real-time tracking of their evolution feasible. After a few theoretical notes on the <formula formulatype="inline"><tex Notation="TeX">$I/Q$</tex></formula> impairments and DEKF, the operative stages of the proposed method are described in detail. Several experiments are then conducted on radio frequency signals to assess the performance of the method; the obtained results are given and discussed. </para>

Computational aspects of continuous–discrete extended Kalman-filtering

This paper elaborates how the time update of the continuous---discrete extended Kalman-filter (EKF) can be computed in the most efficient way. The specific structure of the EKF-moment differential equations leads to a hybrid integration algorithm, featuring a new Taylor---Heun-approximation of the nonlinear vector field and a modified Gauss---Legendre-scheme, generating positive semidefinite solutions for the state error covariance. Furthermore, the order of consistency and stability behavior of the outlined procedure is investigated. The results are incorporated into an algorithm with adaptive controlled step size, assuring a fixed numerical precision with minimal computational effort.

Ultrasonic Time-of-Flight Estimation Through Unscented Kalman Filter

This paper deals with distance or level measurements based on ultrasonic time-of-fight estimation. Moving from a past experience concerning the proposal of a method based on discrete extended Kalman filter (DEKF) to overcome some limitations of already available ultrasonic-based techniques, a new digital signal processing method capable of granting further improvements is presented. The method is based on the unscented Kalman filter (UKF), which is a new extension of the Kalman filter theory mandated to face some DEKF problems, mainly due to its inherent linearization approach. To this aim, UKF is applied to the acquired ultrasonic signal in order to estimate the returned echo envelope as well as to locate its onset more accurately. After describing key features and implementation issues of the new method, the results obtained in a number of tests on simulated and actual ultrasonic signals, which assess its reliability and effectiveness as well as advantages with respect to the previous one, are given.

A Measurement Method Based on Kalman Filtering for Ultrasonic Time-of-Flight Estimation

A new digital signal-processing method for ultrasonic time-of-flight (TOF) estimation is presented. The method applies the discrete extended Kalman filter (DEKF) to the acquired ultrasonic signal in order to accurately estimate the shape factors of the echo envelope as well as locate its onset. It is also possible to assure reduced bias and uncertainty in critical TOF measurements, such as those involving low signal-to-noise ratio (SNR) as well as severe distortion of echo shape. A number of numerical tests are conducted on simulated signals with the aim of highlighting the good performance of the method when operating in critical conditions. Results attained in TOF-based distance measurements finally assess the reliability and efficacy of the method in the presence of actual ultrasonic signals.

Estimation of intracellular phosphate content in plant cell cultures using an extended Kalman filter

Phosphate is an essential nutrient that is usually taken up by plant cells rapidly and stored intracellularly. Currently, there is no convenient means for on-line sensing of the intracellular phosphate content in cultured plant cells. In this study, a state estimator for this important parameter in batch plant cell cultures was developed using extended Kalman filter (EKF) methodology. A non-linear kinetic model was constructed to describe the dynamics of intracellular phosphate uptake and utilization. For intracellular phosphate estimation, this model was found to be most sensitive to three parameters: the maximum specific growth rate (μ max), the maximum phosphate uptake rate (ν max), and the yield coefficient on oxygen ( y o). The EKF algorithm coupled with the kinetic model and on-line oxygen uptake rate ( r o) measurement was used successfully to track the intracellular phosphate content under different initial phosphate concentrations. The state estimator could also accurately predict the biomass concentration. When μ max, ν max, and y o were included in the state vector, tracking of intracellular phosphate was only slightly affected. The estimation system was found very stable as long as the measurement errors of the initial states, the r o measurement error, and the system error were respectively within 30%, 30%, and 50%. With a r o measurement interval as long as 12 h, accurate tracking of the intracellular phosphate content could still be attained using a discrete EKF. Apparently, the slow r o dynamics in plant cell cultures allows the use of a long measurement interval. Considering the difficulties encountered in the on-line sensing of intracellular phosphate using the chemical or nuclear magnetic resonance (NMR) methods, the EKF method coupled with simple on-line oxygen uptake rate measurement provides a promising means for sensing this important culture parameter on-line.