Kalman Filter Formulation Research Articles

Sampling free iterative PCE filter for state and parameter estimation of nonlinear dynamical systems

We present a novel filter for state and parameter estimation in non-linear dynamical systems, based on a generalised Kalman filter formulation. To achieve a sampling-free implementation, polynomial chaos expansion (PCE) and a Galerkin projection method are utilized for the propagation of uncertainties through the system dynamics. The non-linear dynamics of the system are then linearised by a sequence of Gauss-Newton iterations in combination with linear Kalman updates. Additionally, we introduce a new square root implementation of the PCE-based filter. The proposed filter is evaluated on the Lorenz-63 and Lorenz-84 models for the task of simultaneous state and parameter estimation and is compared with two related approaches. Finally, the computational complexity of our square-root implementation is compared against two existing square root approaches.

Invariant Kalman Filter Design for Securing Robust Performance of Magnetic–Inertial Integrated Navigation System under Measurement Uncertainty

This study proposes an enhanced integration algorithm that combines the magnetic field-based positioning system (MPS—Magnetic Pose Estimation System) with an inertial system with the advantage of an invariant filter structure. Specifically, to mitigate the nonlinearity of the propagation model and perturbing effect from the estimated uncertainty, the formulation of the invariant Kalman filter was derived in detail. Then, experiments were conducted to validate the algorithm with an unmanned vehicle equipped with an IMU and MPS receiver. As a result, the navigation performance of the IEKF-based inertial and magnetic field integration system was presented and compared with the conventional Kalman filter results. Furthermore, the convergence and navigation performance were evaluated in the presence of state variable initialization errors. The findings indicate that the inertial and magnetic field coupled with the IEKF outperforms the typical KF approach, particularly when dealing with initial estimate uncertainties.

Monitoring internal heat fluxes on Pulsating Heat Pipes using Kalman filter – Numerical and experimental results

The present study proposes an approach to monitor the wall-to-fluid internal heat flux on Pulsating Heat Pipes (PHP), which provides meaningful information about the efficiency and limiting operation conditions of such devices. In this sense, the current work adopts the Kalman Filter (KF) to estimate the heat flux between the working fluid and the internal tube wall along the adiabatic section of a PHP device by solving an inverse heat conduction problem. The KF is a well-known tracking technique, and therefore the current study is attractive due to the possibility of real-time evaluation of internal heat fluxes, a prospect that is extremely difficult to realize by standard whole domain inverse problem techniques. A high-speed and high-resolution infrared camera performs the temperature acquisition on the external wall of the PHP device. Two mathematical heat conduction models are considered: complete and thin-wall models. The complete model determines the temperature and heat flux evolution over time, and the thin-wall model is used to correct the random walk parameter on the KF formulation. Since the time step needed to the KF evolution model is smaller than the measurements acquisition interval, a matrix manipulation is used during the prediction step, reducing the number of mathematical operations. Hence, instead of predicting the states until reaching the measurement times, a proper mathematical manipulation reduces several mathematical operations to a single one. Then, the KF estimation capability is evaluated by considering synthetic data with different noise levels and different heat flux values. Subsequently, real experimental data are employed, and the results are compared with the ones obtained via a standard whole domain technique. The results obtained with the KF presented good visual agreement with the exact solution. Once the KF presents a delayed response, which is a characteristic of some filtering techniques, the errors with synthetic data were calculated offsetting the delay of the estimation in time in order to make a fair comparison, showing a maximum estimation error lower than 36 % considering the highest noise level. The synthetic data considered 20 s of time experiment, and the KF was able to estimate the PHP internal heat flux in less than 12 s, including the smoothing temperature step, showing fast and accurate results. For the real experimental data, the agreement between the KF and a whole domain technique is highly commendable, demonstrating a deviation of 25.5 % between both techniques. Moreover, the KF required significantly less computational time. The proposed methodology for heat flux estimation on PHPs is promising to evaluate their thermal behavior in real-time due to a low CPU time and computational memory required, as well as its simplicity to be implemented, possibly on embarked devices. Such information may therefore provide a thermal indicator of the correct functioning of PHPs in real time applications.

A sparse matrix formulation of model-based ensemble Kalman filter

We introduce a computationally efficient variant of the model-based ensemble Kalman filter (EnKF). We propose two changes to the original formulation. First, we phrase the setup in terms of precision matrices instead of covariance matrices, and introduce a new prior for the precision matrix which ensures it to be sparse. Second, we propose to split the state vector into several blocks and formulate an approximate updating procedure for each of these blocks. We study in a simulation example the computational speedup and the approximation error resulting from using the proposed approach. The speedup is substantial for high dimensional state vectors, allowing the proposed filter to be run on much larger problems than can be done with the original formulation. In the simulation example the approximation error resulting from using the introduced block updating is negligible compared to the Monte Carlo variability inherent in both the original and the proposed procedures.

Extending the dynamic response of structural input estimation by combining sensors with unmatched bandwidth

The estimation of external loads acting on mechanic structures is of paramount importance for the design, operation, and control of these systems. In many cases, external loads cannot be measured directly due to the intrusive nature or the high costs of adequate sensors. Model-based estimation techniques yield the possibility to estimate external loads in an indirect manner using indirect measurements. These methods usually assume an optimal sensor behaviour over the entire frequency range of the estimator application, which not necessarily has to be the case due to sensor bandwidth limitations. Therefore, model-based sensor fusion is constrained to sensors and applications that share the same bandwidth. This paper proposes a novel non-intrusive approach to estimating external loading in the presence of incongruent measurement regimes of the involved sensors using Kalman filters. The method uses existing model-based estimation techniques to establish a broadly applicable framework for the fusion of multiple sensors with non-coinciding measurement bandwidths. A promising application of this new approach is not only the combination of sensors that measure the same quantity in different frequency regimes but also the combination of sensor types which are not being used together due to different measurement regimes. The newly formulated approach is tested both numerically and with a validation experiment in which accelerometer and gyroscope measurement are combined. In both cases, the new method shows improved behaviour in terms of estimations quality of an unknown force over traditional Kalman filter formulations in the presence of prominent sensor dynamics and over a wide frequency range.

Innovative Formulation in Discrete Kalman Filtering with Constraints - A Generic Framework for Comprehensive Error Analysis

This manuscript establishes a generic framework for comprehensive error analysis in discrete Kalman filtering with constraints, which systematically provides a complete set of algorithmic formulas along with demonstrating an alternative process of theoretical analytics of discrete Kalman filter. This constructive work aims extensively to standardize the formulation of Kalman filter with constraints. In analogy to the similar framework for standard discrete Kalman filter (without any constraints), the proposed framework specifically considers: model formulation vs. the error sources, the solution of the state and process noise vectors, the residuals for the process noise vector and the measurement noise vector, the redundancy contribution of the predicted state vector, process noise vector and measurement vector, and other relevant essential aspects, of which some of the features are essential to comprehensive error analysis, but are nonexistent yet in the primary algorithm in Kalman filtering with constraints. Besides, the algorithmic form of the Extended Kalman filter with constraints is also provided for practical purpose. At the end, specific remarks about the developed framework are given to emphasize on its usage to a certain extend.

Antenna Array-Aided Real-Time Kinematic Using Moving Base Stations

A global navigation satellite systems (GNSS) real-time kinematic (RTK) using multiple base stations is referred as an antenna array-aided RTK. This positioning method more reliably resolves carrier phase integer ambiguities by processing additional measurements provided from multiple GNSS reference stations whose antenna locations are accurately surveyed. In this study, a conventional antenna array-aided RTK is extended to improve RTK performance with multiple moving base stations. Because accurately surveyed antenna coordinates are not available, we propose an antenna array-aided Kalman filter approach to first check the sanity of the fixed integer ambiguities and baselines estimated by moving reference stations and use them as a-priori information in RTK measurement formulations. Our flight test results with three unmanned aerial vehicles confirmed that the proposed Kalman filter formulation outperformed the conventional RTK and a priori antenna array-aided RTK approaches in the aspects of positioning accuracy, time to first fix, and success probability of integer ambiguity fixes.

Recursive estimation of the stochastic model based on the Kalman filter formulation

Based on the batch expectation–maximization (EM) and recursive least-squares algorithms, we develop a new recursive variance components estimation (Recursive-VCE) algorithm that applies a Kalman filter and validates it by a simulated kinematic precise point positioning (PPP) experiment and a PPP test on real-world data. The Recursive-VCE algorithm processes the observations in an epoch-by-epoch or a group-by-group manner. Once new observations are obtained, it updates the estimates of the variance components in a recursive way or on the fly. Therefore, it does not require significant computing resources to store sufficiently large training datasets. The resulting algorithm is simple and able to be easily adapted to determine time-varying behaviours and is shown to converge faster than the batch EM algorithm because the EM algorithm updates the parameters only once after dealing with all the data. Hence, it is a good complement to other batch VCE methods, and its application in real-time data processing is promising.

A Novel Kalman Filter Formulation for Improving Tracking Performance of the Extended Kernel RLS

In this paper, two contributions are presented. Firstly, a novel variant of the Kalman filter is developed and used to improve the tracking performance of the extended kernel recursive least squares algorithm (Ex-KRLS). Without resorting to the Riccati equation, the proposed formulation of the Kalman filter relies on principles of optimization and convex duality to obtain a stable quadratic form. It allows for performance and stability efficiency by minimizing the estimation error with the manipulation of a single scalar variable. Secondly, the proposed Kalman filter formulation is embedded into the Ex-KRLS-KF algorithm, an extension of the Ex-KRLS, to further improve its tracking performance. For this purpose, the state model of the proposed Ex-KRLS-KF variant is constructed in the original state space, while the hidden state is then estimated using the proposed Kalman filter formulation. The standard KRLS algorithm learns the measurement model used in hidden state estimation. We show through a comprehensive set of computer experiments that the proposed hybrid algorithm has more flexible state and noise models than competing alternative algorithms. We evaluate the proposed algorithm in two benchmarking tasks (nonlinear Rayleigh multipath channel tracking and Lorenz system modeling) and compare its performance with those provided by the state-of-the-art algorithms.

Recursive estimation for sparse Gaussian process regression

Gaussian Processes (GPs) are powerful kernelized methods for non-parametric regression used in many applications. However, their use is limited to a few thousand of training samples due to their cubic time complexity. In order to scale GPs to larger datasets, several sparse approximations based on so-called inducing points have been proposed in the literature. In this work we investigate the connection between a general class of sparse inducing point GP regression methods and Bayesian recursive estimation which enables Kalman Filter like updating for online learning. The majority of previous work has focused on the batch setting, in particular for learning the model parameters and the position of the inducing points, here instead we focus on training with mini-batches. By exploiting the Kalman filter formulation, we propose a novel approach that estimates such parameters by recursively propagating the analytical gradients of the posterior over mini-batches of the data. Compared to state of the art methods, our method keeps analytic updates for the mean and covariance of the posterior, thus reducing drastically the size of the optimization problem. We show that our method achieves faster convergence and superior performance compared to state of the art sequential Gaussian Process regression on synthetic GP as well as real-world data with up to a million of data samples.

Real‐Time Earthquake Location Based on the Kalman Filter Formulation

AbstractSeismic location is an essential task for earthquake monitoring. The general practice is to locate earthquakes using arrival times from all recorded stations. However, this is not well suited for real‐time applications such as the earthquake early warning, where earthquake locations need to be determined and updated as more stations are triggered. Here we have developed a real‐time linear location method based on a Kalman filter formulation. It updates location and its uncertainty whenever a station is triggered. We have demonstrated its effectiveness with synthetic and real data sets in Parkfield, California in a retrospective mode. The tests show that we can obtain relatively accurate locations and reliable uncertainties for earthquakes with 4 or 5 stations triggered. In particular, without considering the station latency, accurate earthquake locations can be achieved retrospectively in about 3.3 and 2.7 s for the 2003 San Simeon Mw6.6 earthquake and 2004 Parkfield Mw6.0 earthquake, respectively.

Identification of abrupt stiffness changes of structures with tuned mass dampers under sudden events

This paper presents a recursive system identification method for multi-degree-of-freedom (MDoF) structures with tuned mass dampers (TMDs) considering abrupt stiffness changes in case of sudden events, such as earthquakes. Due to supplementary non-classical damping of the TMDs, the system identification of MDoF+TMD systems disposes a challenge, in particular, in case of sudden events. This identification methods may be helpful for structural health monitoring of MDoF structures controlled by TMDs. A new adaptation formulation of the unscented Kalman filter allows the identification method to track abrupt stiffness changes. The paper, firstly, describes the theoretical background of the proposed system identification method and afterwards presents three parametric studies regarding the performance of the method. The first study shows the augmented state identification by the presented system identification method applied on a MDoF+TMD system. In this study, the abrupt stiffness changes of the system are successfully detected and localized under earthquake, impulse and white noise excitations. The second study investigates the effects of the state covariance and its relevance for the system identification of MDoF+TMD systems. The results of this study show the necessity of an adaptive definition of the state covariance as applied in the proposed method. The third study investigates the effects of modeling on the performance of the identification method. Mathematical models with discretization of different orders of convergence and system noise levels are studied. The results show that, in particular, MDoF+TMD systems require higher order mathematical models for an accurate identification of abrupt changes.

State estimation for heavy oil hydroprocessing reactors using extended Kalman filters

A state estimator for a class of slurry-phase reactors for heavy oil hydroprocessing is presented. The algorithm is based on the extended Kalman filter formulation for non-linear systems under plant-model mismatch. Although state and parameter estimation have been widely used in the literature for different reacting systems, so far it has not been applied in multiphase slurry reactors for heavy oil hydroprocessing since modeling and simulation of this type of systems is still an emerging area. The reactor and kinetic models used to formulate the filter algorithm were taken from the literature for hydrocracking and hydrotreating reactions of an atmospheric residue (312 °C+) for different slurry-phase reactor configurations. In order to simulate a real plant operation were measurements for heavy oil composition are difficult to obtain online, the present study assumed that the system states were estimated using only temperature measurements and the proposed slurry-phase reactor models. Simulation results have shown that the extended Kalman filter provides acceptable estimations of the systems states and that they can be used for online control and process optimization. However, these results need to be validated once experimental data for this system becomes available.

Leveraging Localization Accuracy With Off-Centered GPS

A GPS is usually installed at the center of a vehicle because with this configuration its data become the position of the vehicle without complex calibration and transformation. In this paper, however, we verify that the off-center arrangement of the GPS improves the position and orientation accuracy theoretically and experimentally. The extended Kalman filter (EKF) formulation and its observability and uncertainty analysis present that the off-centered GPS is better observable and less uncertain. Moreover, our experiments with synthetic data and real sensor dataset confirm the improvement of localization accuracy. The synthetic data generated in various situations reveal the important properties of localization with the off-centered GPS. The public outdoor dataset with real GPS data also supports the advantage of the off-centered GPS.

Parameter estimation for externally simulated thermal network models

Obtaining accurate dynamic models of building thermal behaviour requires a statistically solid foundation for estimating unknown parameters. This is especially important for thermal network grey-box models, since all their parameters normally need to be estimated from data. One attractive solution is to maximise the likelihood function, under the assumption of Gaussian distributed residuals. This technique was developed previously and implemented in the Continuous Time Stochastic Modelling framework, where an Extended Kalman Filter is used to compute residuals and their covariances. The main result of this paper is a similar method applied to a thermal network grey-box model of a building, simulated as an electric circuit in an external tool. The model is described as a list of interconnected components without deriving explicit equations. Since this model implementation is not differentiable, an alternative Kalman filter formulation is needed. The Unscented and Ensemble Kalman Filters are designed to handle non-linear models without using Jacobians, and can therefore also be used with models in a non-differentiable form. Both Kalman filter implementations are tested and compared with respect to estimation accuracy and computation time. The Profile Likelihood method is used to analyse structural and practical parameter identifiability. This method is extended to compute two-dimensional profiles, which can also be used to analyse parameter interdependence by providing insight into the parameter space topology.

New combination of super-twisting and Kalman filter for direct power control of distributed generation at different operations

This paper presents an efficient modification of the direct power control scheme employed to precisely control the power of the distributed generation system. This modification is articulated for utilizing the super-twisting technique for controlling the feedforward power and Kalman filter for estimating the feedback power. The combination of the super-twisting technique and Kalman filter for the direct power control achieves a seamless steady-state and dynamic performance for the injected power by the distributed generation system in a grid-connected mode. Furthermore, the concept of direct power control is modified to operate the distributed generation system in a stand-alone mode so as to stabilize the voltage at different loading conditions, where the feedback voltage is estimated by a simple formulation of Kalman filter. The proposed control scheme for the distributed generation system is compared to several control schemes such as the conventional direct power control operated by space vector modulation and another proposed 1st order sliding mode control based direct power scheme. Simulation results are presented to prove the meritorious performance of Kalman filter and the super-twisting technique for the power and voltage control schemes. Experimental results are provided to prove the practicality of the super-twisting technique for the operation of the direct power control in a grid-connected mode.

Magneto-Inertial Integrated Navigation System Design Incorporating Mapping and Localization Using Concurrent AC Magnetic Measurements

A new artificially generated magnetic-field-based alternative navigation technique is investigated for achieving seamless navigation in non-line-of-sight (NLOS) environments where satellite navigation is unavailable, and where vision, lidar, and other sensors that make use of signals of opportunity cannot provide reliable estimation. Unlike the previously reported multilateration methods, which make use of the magnetic receiver coil's voltage model, the proposed method first extracts three-dimensional field vectors from concurrent ac magnetometer measurements, then establishes the spatial representation model of magnetic vectors for localization purpose. It does so by separating each frequency component and developing a geometric matching procedure using the interpolated data from the grid map in order to achieve a pose estimation. This study further presents an integrated navigation algorithm that associates an inertial measurement with the estimated pose information to secure a continuous navigation solution at a high output rate, where the extended Kalman filter (KF) formulation is typically employed. Finally, in order to validate the estimation performance of the proposed method, a small-scale trajectory experiment in 70cm × 70cm × 50 cm test platform was performed, by which the integrated navigation was verified as possessing an average position accuracy in the sub-cm range as well as a heading angle error in the sub-degree range.

Angular Rate and Alignment Estimation for Gyroless Spacecraft by Only Star Trackers

An extended Kalman filtering technique is addressed for gyroless spacecraft to estimate the angular rate and attitude vectors of spacecraft and the alignment of reaction wheels using only vector measurement. In this work, it is assumed that the angular momentum and control torque of the reaction wheels are measurable and usable for the filter formulation. Because there is no angular rate sensor, a dynamic model of a spacecraft equipped with reaction wheels is required to estimate the angular rate of the spacecraft. In this formulation, misalignment of the reaction wheels is considered a state variable to be estimated. For actual implementation, the external torque disturbance is regarded as a first-order Markov process, and is also exploited in this Kalman filter formulation. Finally, effectiveness of the estimation algorithm proposed in this paper is demonstrated using numerical simulations.

Adaptive robust cubature Kalman filtering for satellite attitude estimation

This paper is concerned with the adaptive robust cubature Kalman filtering problem for the case that the dynamics model error and the measurement model error exist simultaneously in the satellite attitude estimation system. By using Hubel-based robust filtering methodology to correct the measurement covariance formulation of cubature Kalman filter, the proposed filtering algorithm could effectively suppress the measurement model error. To further enhance this effect and reduce the impact of the dynamics model error, two different adaptively robust filtering algorithms, one with the optimal adaptive factor based on the estimated covariance matrix of the predicted residuals and the other with multiple fading factors based on strong tracking algorithm, are developed and applied for the satellite attitude estimation. The quaternion is employed to represent the global attitude parameter, and three-dimensional generalized Rodrigues parameters are introduced to define the local attitude error. A multiplicative quaternion error is derived from the local attitude error to maintain quaternion normalization constraint in the filter. Simulation results indicate that the proposed novel algorithm could exhibit higher accuracy and faster convergence compared with the multiplicative extended Kalman filter, the unscented quaternion estimator, and the adaptive robust unscented Kalman filter.

Monthly Forecasting of the Dollar to the Ruble Exchange Rate with Adaptive Kalman Filter

The goal: to develop a model that allows you to forecast the dollar to the ruble exchange rate for a month ahead based on macroeconomic data, published at monthly intervals. Proposed structural model of the dynamics of the ruble and dollar masses that determine the exchange rate, depending on changes in foreign exchange reserves, the balance of foreign trade, the monetary base, the MICEX index, the price of oil. With the help of the Kalman filter (KF), the model parameters, the dynamics of the money masses were estimated, and forecasting of the dollar exchange rate was done. Monthly data were used from the beginning of 2015 to mid-2017. The estimation of the capacity of dollar market was found in about half the capacity of the MICEX index funds. Average error of forecasts, based on information available one step before the forecasted moments (RMSEA) was 1.99. Adaptive form of KF was developed when, similarly to the EM algorithm, the phases of KF estimation in the window and minimization of average prediction error to determine the optimal estimates for the system model parameters in this moment are sequentially alternated. With this RMSEA became 1.39.