This book presents a thorough discussion of the mathematical theory and computational schemes of Kalman filtering. The filtering algorithms are derived via different approaches, including a direct ...method and an indirect method.
In the previous investigation, we demonstrated that the augmented extended Kalman filter (AEKF) can effectively address the limitations of the conventional extended Kalman filter (EKF) in denoising ...electrocardiogram (ECG) signals contaminated with nonstationary noise. The AEKF employs a novel measurement model designed to accommodate non-Gaussian, non-stationary additive disturbances. However, its performance relies on careful tuning of certain hyperparameters. In this study, we introduce an advanced version of the AEKF, referred to as the adaptive dual augmented extended Kalman filter (ADAEKF), which features an automated mechanism for optimizing hyperparameter settings. Furthermore, we use a methodology to adaptively adjust state and measurement noise covariance matrices, leading to substantial improvements in ADAEKF’s performance. Unlike the original AEKF, which assumed nonstationary noise in ECG signals followed an autoregressive order one (AR1), our proposal allows for the possibility of second-order (AR2) or third-order (AR3) processes. We develop alternative measurement models based on these hypotheses. Notably, the ADAEKF does not require prior knowledge of the specific noise type and can effectively handle white Gaussian processes as well. Our work represents the first effort to integrate and assess the dual EKF framework within the domain of ECG signal processing. Extensive evaluations conducted on the MIT-BIH Normal Sinus Rhythm Database (NSRDB) using three distinct types of contaminants – white Gaussian noise, synthetic pink noise, and real muscle artifact noise obtained from the Physionet noise stress database – demonstrate the superiority of the proposed ADAEKF over traditional EKF and AEKF approaches in both stationary and non-stationary environments. Furthermore, the proposed method was benchmarked against other nonlinear Kalman filter-based frameworks, including the ensemble Kalman filter (EnKF), unscented Kalman filter (UKF), and the recently introduced adaptive cubature Kalman filter (ACKF) for ECG denoising. The comparative results revealed that the proposed ADAEKF outperformed UKF, ACKF, and EnKF in nonstationary environments at low input SNRs. Additionally, the proposed ADAEKF improved the overall performance of EKF-based frameworks compared to the state-of-the-art marginalized particle extended Kalman filter (MP-EKF), while offering a simpler and more computationally efficient solution.
Data Assimilation comprehensively covers data assimilation and inverse methods, including both traditional state estimation and parameter estimation. This text and reference focuses on various ...popular data assimilation methods, such as weak and strong constraint variational methods and ensemble filters and smoothers. It is demonstrated how the different methods can be derived from a common theoretical basis, as well as how they differ and/or are related to each other, and which properties characterize them, using several examples. It presents the mathematical framework and derivations in a way which is common for any discipline where dynamics is merged with measurements. The mathematics level is modest, although it requires knowledge of basic spatial statistics, Bayesian statistics, and calculus of variations. Readers will also appreciate the introduction to the mathematical methods used and detailed derivations, which should be easy to follow, are given throughout the book. The codes used in several of the data assimilation experiments are available on a web page. The focus on ensemble methods, such as the ensemble Kalman filter and smoother, also makes it a solid reference to the derivation, implementation and application of such techniques. Much new material, in particular related to the formulation and solution of combined parameter and state estimation problems and the general properties of the ensemble algorithms, is available here for the first time. The 2nd edition includes a partial rewrite of Chapters 13 an 14, and the Appendix. In addition, there is a completely new Chapter on 'Spurious correlations, localization and inflation', and an updated and improved sampling discussion in Chap 11.
This article proposes two Kalman-filter-based online identification schemes for permanent magnet synchronous machines (PMSMs), where the nonlinearity of a voltage-source inverter (VSI) is taken into ...account. One is formulated from an extended Kalman filter; the other uses a dual extended Kalman filter. They are generally formulated and can be applied to any identifiable electrical parameter combinations. The proposed schemes are further implemented on an industrial embedded control system. Their performance tests are conducted on a PMSM under static and dynamic conditions and compared with the extended Kalman filter without VSI nonlinearity compensation. The effectiveness of the proposed approaches is proved by the experimental results. Furthermore, a sensitivity analysis of the initial setup of parameter estimates has shown that the proposed estimators are robust against poor initial value choices. Real-time feasibility of the proposed estimators up to <inline-formula><tex-math notation="LaTeX">\text{20}\;\text{kHz}</tex-math></inline-formula> is demonstrated via experiments.
This paper elaborates the Accurate Continuous-Discrete Extended Kalman Filter grounded in an ODE solver with global error control and its comparison to the Continuous-Discrete Cubature and Unscented ...Kalman Filters. All these state estimators are 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. Our numerical results show that all the methods can be used for practical target tracking, but the Accurate Continuous-Discrete Extended Kalman Filter is more flexible and robust. It treats successfully (and without any manual tuning) the air traffic control scenario for various initial data and for a range of sampling times.
Accurate information about dynamic states is important for efficient control and operation of a power system. This paper compares the performance of four Bayesian-based filtering approaches in ...estimating dynamic states of a synchronous machine using phasor measurement unit data. The four methods are extended Kalman filter, unscented Kalman filter, ensemble Kalman filter, and particle filter. The statistical performance of each algorithm is compared using Monte Carlo methods and a two-area-four-machine test system. Under the statistical framework, robustness against measurement noise and process noise, sensitivity to sampling interval, and computation time are evaluated and compared for each approach. Based on the comparison, this paper makes some recommendations for the proper use of the methods.
This paper presents a continuous control set model-predictive control with a receding horizon for a three-phase voltage-source inverter with an LCL filter. In this proposal, a reduced model of the ...converter with an embedded integrator and a Kalman filter are used to obtain the inverter-side currents without oscillation. Then, a cost function makes use of these currents to generate the optimum duty cycles for the space vector modulator. With the proposed method, active damping performances, a zero state error, and a reduction of the computational burden are achieved. Compared to the finite control set model-predictive control, the proposed method operates with a fixed switching frequency. Simulations and experimental results show that this proposal works correctly even in the case of grid harmonics and voltage sags.
This paper presents a visual-inertial odometry framework that tightly fuses inertial measurements with visual data from one or more cameras, by means of an iterated extended Kalman filter. By ...employing image patches as landmark descriptors, a photometric error is derived, which is directly integrated as an innovation term in the filter update step. Consequently, the data association is an inherent part of the estimation process and no additional feature extraction or matching processes are required. Furthermore, it enables the tracking of noncorner-shaped features, such as lines, and thereby increases the set of possible landmarks. The filter state is formulated in a fully robocentric fashion, which reduces errors related to nonlinearities. This also includes partitioning of a landmark’s location estimate into a bearing vector and distance and thereby allows an undelayed initialization of landmarks. Overall, this results in a compact approach, which exhibits a high level of robustness with respect to low scene texture and motion blur. Furthermore, there is no time-consuming initialization procedure and pose estimates are available starting at the second image frame. We test the filter on different real datasets and compare it with other state-of-the-art visual-inertial frameworks. Experimental results show that robust localization with high accuracy can be achieved with this filter-based framework.
•The surface heat flux and liquid-phase interface can be retrieved in real time.•A non-intrusive measurement technique is applied in the present study.•Comparing with EKF and UKF, the KF can only be ...used to solve linear problems.•Compared to UKF, the EKF can only be applied to solve weak nonlinear problem.
The non-intrusive inverse heat transfer technique based on the Kalman filtering (KF) method is proposed for on-line retrieving the time-dependent boundary heat flux, internal temperature distribution, and liquid-phase interface of the participating medium simultaneously. To obtain the measured temperature signals, the nonlinear conduction-radiation heat transfer with phase change in the participating medium is solved by the enthalpy method combined with discrete ordinate method. Three different types of retrieval algorithms have been employed: the original Kalman filter (KF), the extended KF (EKF), and the unscented KF (UKF). The ideal participating media, which is assumed to be anisotropic scattering, and opaque and diffuse gray boundary, is employed to verify the reliability and validity of the proposed algorithm. Comparing with the EKF and UKF, the original KF cannot be applied to solve the nonlinear problems. Comparing with the UKF, the EKF can only be employed to solve the weak nonlinear problem. For the inverse coupled conduction-radiation problem with phase change, the UKF is proved to be more robust to estimate the time-dependent heat flux, internal temperature distribution, and liquid-phase interface simultaneously in real time.
In this paper, in order to enhance the numerical stability of the unscented Kalman filter (UKF) used for power system dynamic state estimation, a new UKF with guaranteed positive semidifinite ...estimation error covariance (UKF-GPS) is proposed and compared with five existing approaches, including UKF-schol, UKF-<inline-formula> <tex-math notation="LaTeX">\boldsymbol {\kappa } </tex-math></inline-formula>, UKF-modified, UKF-<inline-formula> <tex-math notation="LaTeX">\boldsymbol {\Delta Q} </tex-math></inline-formula>, and the square-root UKF (SR-UKF). These methods and the extended Kalman filter (EKF) are tested by performing dynamic state estimation on WSCC 3-machine 9-bus system and NPCC 48-machine 140-bus system. For WSCC system, all methods obtain good estimates. However, for NPCC system, both EKF and the classic UKF fail. It is found that UKF-schol, UKF-<inline-formula> <tex-math notation="LaTeX">\boldsymbol {\kappa } </tex-math></inline-formula>, and UKF-<inline-formula> <tex-math notation="LaTeX">\boldsymbol {\Delta Q} </tex-math></inline-formula> do not work well in some estimations while UKF-GPS works well in most cases. UKF-modified and SR-UKF can always work well, indicating their better scalability mainly due to the enhanced numerical stability.
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