Set Of Sigma Points Research Articles

An unscented Kalman filter under unknown input without direct feedthrough for joint input and system identification of structural systems

Given the difficulty of accurately measuring external inputs acting on structures, the research on joint input-system identification has gained significant attention in recent years. Among them, several unscented Kalman filters under unknown input (UKF-UI) based methods have been proposed for joint input and system identification. However, the existing UKF-UI methods require unknown inputs to be presented in the observation equations, which is not always available in practical engineering. To solve this problem, an unscented Kalman filter under unknown input without direct feedthrough (UKF-UI-WDF) is proposed in this paper for real-time joint input and system identification of structural systems without direct feedthrough using limited response measurements. The proposed UKF-UI-WDF is derived for recursive identification of the structural states, parameters, and unknown inputs by integrating unknown input estimation into the conventional UKF framework. In each identification step, the unknown inputs are first identified by minimizing posterior residuals of measurements, and then the structural states and parameters are identified by a traditional unscented Kalman filtering using the identified unknown inputs. According to whether the uncertainty of the identified unknown inputs can be quantitatively described or not, two implementation algorithms are provided in this paper. In algorithm 1, only the mean values of unknown inputs can be identified. In algorithm 2, the unknown inputs are identified as a set of sigma points, so not only the mean values but also the variance of the unknown inputs can be estimated. The effectiveness of the proposed method is examined by the joint input and system identification of various case studies of linear and nonlinear hysteretic shear structures subjected to unknown external or base excitations, respectively. The results show that the proposed UKF-UI-WDF method can effectively perform real-time joint input-system identification of structural systems without direct feedthrough using limited response measurements.

Improved Compound Correction‐Electrical Equivalent Circuit Modeling and Double Transform‐Unscented Kalman Filtering for the High‐Accuracy Closed‐Circuit voltage and State‐of‐Charge Co‐Estimation of Whole‐Life‐Cycle Lithium‐Ion batteries

For complex energy storage conditions, it is necessary to monitor the state‐of‐charge (SOC) and closed‐circuit voltage (CCV) status accurately for the reliable power supply application of lithium‐ion batteries. Herein, an improved compound correction‐electrical equivalent circuit modeling (CC‐EECM) method is proposed by considering the influencing effects of ambient temperature and charge–discharge current rate variations to estimate the CCV. Then, an improved adaptive double transform‐unscented Kalman filtering (ADT‐UKF) method is constructed with recursive sampling data correction to estimate the nonlinear SOC. A dynamic window function filtering strategy is constructed to obtain the new sigma point set for the online weighting coefficient correction. For a temperature range of 5–45 °C, the CCV for the improved CC‐EECM responds well with a maximum error of 0.008608 V, and the maximum SOC estimation error is 6.317%. The proposed ADT‐UKF method improves the CCV and SOC estimation reliability and adaptability to the time‐varying current rate, temperature, and aging factors.

Stochastic Optimal Control of Economic Growth Model under Research and Development Investment with Kalman Filtering Approaches

This article examines an economic growth model that expresses the interaction between production, technology stock, and research and development (R&D) investments. The goal of this study is to maximize production. Considering the presence of Gaussian white noises, this model is reformulated as a stochastic optimal control problem, where the R&D investment rate is defined as the control input. We aim to explore the efficiency of Kalman filtering approaches for solving this problem. Here, the extended Kalman filter (EKF) and unscented Kalman filter (UKF) are applied for state estimation. The state equation linearization is made in the EKF, while the unscented transform is taken in the UKF for generating a set of sigma points. These approaches aim to estimate the state dynamics from different perspectives. With these state estimates, two different computational algorithms are proposed, the EKF for state-control (EKF4SC) and UKF for state-control (UKF4SC) algorithms. The optimal control policy is designed to minimize the cost function. For illustration, the model's parameters are considered in the simulation experiment. The simulation results showed that the UKF has higher accuracy in the state estimation with the smallest mean squares of error compared with the EKF. Moreover, the optimal control policy based on the state estimate generated from the UKF could optimize the cost function of the problem. Hence, the results show the efficiency of the algorithms proposed. In conclusion, the application of the Kalman filtering algorithms to the economic growth model is presented. The significance of this study is to provide a stochastic optimal control model for the economic growth problem and to suggest an efficient computational approach for solving this stochastic optimal control problem.

Implementation and testing of unscented transformation and low rank approximation to enhance SCALE code uncertainty calculations

Recently, the SCALE6.2 code has implemented the Sampler module for randomly sampling important reactor physics inputs from their parental distributions. The Sampler module uses a stochastic sampling technique based on multivariate (MV) sampling which assumes that the input cross-sections are correlated according to a predefined covariance matrix. Although the methodology used in the Sampler module is relatively straightforward and robust, it can be computationally expensive for applications where a large number of Monte Carlo samples are required. In this paper, the SCALE6.2 capabilities are enhanced by implementing the Unscented Transformation (UT) and Low Rank Approximation (LRA) to decrease computational time. The uncertainties are calculated for k-inf due to perturbations in the multi-group cross-sections for a standard PWR fuel pin but the methodology can be applied to any output parameter of interest (such as few group cross sections). First, the UT is applied to generate a set of sigma points which represent the behavior of the system over the entire space. However, since the input model dimensionality is very high, due to the number of isotopes, reactions, and energy-groups, the generation of the sigma sample points may be computationally burdensome. Therefore, the LRA is applied to reduce the order of sampling space which can significantly reduce the computational cost; this model is called the Truncated Unscented Transformation (TUT). The results show that the input-dimension can be reduced from 896 to 112-subspace and the uncertainties in k-inf calculated by TUT sampling with 225 samples are about 5.62×10-1 which agrees well with the full MV sampling with N = 1000 runs and results from TSUNAMI. The results also indicate that the uncertainties calculated by the TUT methodology are not sensitive to the scaling parameter λ assumed in the LRA as long as the appropriate weights are applied.

A Local Unscented Transform Kalman Filter for Nonlinear Systems

AbstractThis paper proposes an efficient data assimilation approach based on the sigma-point Kalman filter (SPKF). With a potential for nonlinear filtering applications, the proposed approach, designated as the local unscented transform Kalman filter (LUTKF), is similar to the SPKF in that the mean and covariance of the nonlinear system are estimated by propagating a set of sigma points—also referred to as ensemble members—generated using the scaled unscented transformation (SUT), while making no assumptions with regard to nonlinear models. However, unlike the SPKF, the LUTKF can reduce the influence of observations on distant state variables by employing a localization scheme to suppress spurious correlations between distant locations in the error covariance matrix. Moreover, while the SPKF uses the augmented state vector constructed by concatenating the model states, model noise, and measurement noise, the system state for the LUTKF is not augmented with the random noise variables, thereby providing an accurate state estimate with relatively few sigma points. In sensitivity experiments executed with a 40-variable Lorenz system, the LUTKF required only three sigma points to prevent filter divergence for linear/nonlinear measurement models. Comparisons of the LUTKF and the local ensemble transform Kalman filters (LETKFs) reveal the advantages of the proposed filter in situations that share common features with geophysical data assimilation applications. In particular, the LUTKF shows considerable benefits over LETKFs when assimilating densely spaced observations that are related nonlinearly to the model state and that have high noise levels—such as the assimilation of remotely sensed data from satellites and radars.

Implementation of the unscented transformation with low rank approximation in uncertainty analysis during large-break loss of coolant accident

The Low Rank Approximation (LRA) and Unscented Transform (UT) are integrated to produce a new algorithm having the capability to decrease the time required for the uncertainty quantification during Loss of coolant accident (LOCA) in Pressurized Water Reactors (PWR). The LRA is an efficient technique used in reducing computational cost due to its ability to perform dimensionality reduction by revealing the active or important degrees of freedom and calculate the basis of the so-called active subspace basing on the Singular Value Decomposition (SVD). For further reduction in the computational time; the UT algorithm is also implemented to generate a set of sigma points, these sigma points are the representatives of the whole probability distribution (the UT is restricted to Gaussian distribution). The main safety parameter is the maximum cladding temperatures during the accident which are computed by ATHLET thermal-hydraulic code. The reactivity coefficients and the covariance matrix are calculated using the SCALE 6.2 code. The present calculation model has 14-dimensions, therefore the number of sigma points needed for the SVD/UT technique is 29, and can be minimized to 5 sigma points only if the LRA/UT is used where two singular values are sufficient to reproduce/span the space thanks to the strong correlations between the reactivity coefficients.

Dynamic state estimation of generators using spherical simplex unscented transform‐based unbiased minimum variance filter

Dynamic state estimation is essential in case of various monitoring, control, and protection strategies that are designed based on the state-space model. Kalman filter-based estimation algorithms are mainly used to estimate these states locally using the input and output measurements of the generator. However, in the case of wide-area power system control and protection strategies, remote estimation of these states is required. This remote estimation relies upon phasor measurement units for measurement signals, which are limited to output measurements such as voltage, current, and frequency. For Kalman filter-based techniques, apart from the output, input measurements such as field and torque input are also required to estimate the states. This study proposes an input invariant filter technique using unbiased minimum variance filter and spherical simplex unscented transform for remote estimation of generator states using the limited phasor measurement unit measurements. The estimation is performed in the absence of mechanical input torque and field voltage measurements using a minimum set of sigma points. The performance of the filter under various transient conditions and in the presence of measurement errors are analysed and compared with existing techniques.

Efficient uncertainty quantification for PWR during LOCA using unscented transform with singular value decomposition

This paper discusses one of the most important issues facing the regulatory body while performing the uncertainty analysis of the nuclear reactor parameter during accident conditions. This problem is the long computational time required by the statistical sampling methods to compute the uncertainty. We overcome this problem by introducing the Unscented Transform (UT) algorithm and singular value decomposition (SVD). Where both algorithms are combined (SVD/UT) to generate a set of sigma points, these sigma points are the representatives of whole probability distribution. The uncertainty quantification is performed during Loss of coolant accident in Pressurized Water Reactor (PWR), where the input variable of uncertainty is the coolant density reactivity. The SCALE 6.2 code is used for calculating the reactivity coefficients and the covariance matrix. The response variables are the peak cladding temperatures during the accident which are computed by ATHLET thermal-hydraulic code. The results obviously confirm the efficiency of the SVD/UT sampling in predicting the new mean values, and assure its ability to reduce the sampling size leading to a dramatic reduction of computational cost.

Navigation Performance Enhancement Using IMM Filtering for Time Varying Satellite Signal Quality

A novel scheme using fuzzy logic based interacting multiple model (IMM) unscented Kalman filter (UKF) is employed in which the Fuzzy Logic Adaptive System (FLAS) is utilized to address uncertainty of measurement noise, especially for the outlier types of multipath errors for the Global Positioning System (GPS) navigation processing. Multipath is known to be one of the dominant error sources, and multipath mitigation is crucial for improvement of the positioning accuracy. It is not an easy task to establish precise statistical characteristics of measurement noise in practical engineering applications. Based on the filter structural adaptation, the IMM nonlinear filtering provides an alternative for designing the adaptive filter in the GPS navigation processing for time varying satellite signal quality. The uncertainty of the noise can be described by a set of switching models using the multiple model estimation. An UKF employs a set of sigma points by deterministic sampling, which avoids the error caused by linearization as in an extended Kalman filter (EKF). For enhancing further system flexibility, the fuzzy logic system is introduced. The use of IMM with FLAS enables tuning of appropriate values for the measurement noise covariance so as to obtain improved estimation accuracy. Performance assessment will be carried out to show the effectiveness of the proposed approach for positioning improvement in GPS navigation processing.

Effective Computational Approach for Prediction and Estimation of Space Object Breakup Dispersion during Uncontrolled Reentry

This paper provides an effective approach for the prediction and estimation of space debris due to a vehicle breakup during uncontrolled reentry. For an advanced analysis of the time evolution of space debris dispersion, new efficient computational approaches are proposed. A time evolution of the dispersion of space pieces from a breakup event to the ground impact time is represented in terms of covariance ellipsoids, and in this paper, two covariance propagation methods are introduced. First, a derivative-free statistical linear regression method using the unscented transformation is utilized for performing a covariance propagation. Second, a novel Gaussian moment-matching method is proposed to compute the estimation of the covariance of a debris dispersion by using a Gauss-Hermite cubature-based numerical integration approach. Compared to a linearized covariance propagation method such as the Lyapunov covariance equation, the newly proposed Gauss-Hermite cubature-based covariance computation approach could provide high flexibilities in terms of effectively representing an initial debris dispersion and also precisely computing the time evolution of the covariance matrices by utilizing a larger set of sigma points representing debris components. In addition, we also carry out a parametric study in order to analyze the effects on the accuracy of the covariance propagation due to modeling uncertainties. The effectiveness of the newly proposed statistical linear regression method and the Gauss-Hermite computational approach is demonstrated by carrying out various simulations.

Distributed estimation for stochastic hybrid systems with event‐triggered sensor schedule

This study deals with the state estimation problem for stochastic hybrid systems with event-triggered sensor schedule. By employing the basic interacting multiple model (IMM) approach, a novel event-triggered state estimator has been proposed for stochastic hybrid systems based on a closed-loop schedule rule. To compute the mode probabilities in the IMM estimator, a set of sigma points are generated to produce the pseudo-measurements when the sensor measurements are unavailable. Then, the event-triggered state estimator is extended to develop a distributed estimator in the sense of linear minimum mean square error, where the fused estimates are used to implement the re-initialisation in the interacting stage of the IMM estimator. The performance of proposed estimators is illustrated through the Monte Carlo simulations involving tracking a manoeuvring target in the two-dimensional experiment. Simulation results show that the performance of the proposed estimator is compared with that of the existing filter with a reduced computational burden.

New minimum sigma set for unscented filtering

SummaryIn this work, a new minimum set of sigma points for unscented filtering is proposed along with its unscented Kalman filter in both square‐root and nonsquare‐root forms. Comparative with the other reduced sigma sets of the literature, the new sigma set is, in some cases, better defined or, in another case, a generalization. In numerical examples, the unscented transform of the new sigma set is compared with the unscented transforms of the other reduced sigma sets of the literature. In addition, the performance of the new unscented Kalman filter is studied in an aircraft target tracking scenario. Copyright © 2014 John Wiley & Sons, Ltd.

Sigma-Point Set Rotation in Unscented Kalman Filter: Analysis and Adaptation

The paper deals with analysis and illustration of the impact of the σ-point set rotation on the approximation quality of the unscented transformation and the estimation performance of the unscented Kalman filter. It is shown that the covariance matrix factor, used in σ-point computation, can be multiplied by an arbitrary rotation matrix which moves the σ-points along the surface of a hyper-ellipsoid related to the covariance matrix. The rotation matrix can be thus considered as another user-defined parameter (in addition to the scaling parameter) and the unscented Kalman filter with adaptive selection of both user-defined parameters is proposed. The impact of fixed or adaptively selected parameters on the performance of the unscented Kalman filter is illustrated by a numerical study.

Nonlinear Filtering with IMM Algorithm for Ultra-Tight GPS/INS Integration

Abstract This paper conducts a performance evaluation for the ultra-tight integration of a Global positioning system (GPS) and an inertial navigation system (INS), using nonlinear filtering approaches with an interacting multiple model (IMM) algorithm. An ultra-tight GPS/INS architecture involves the integration of in-phase and quadrature components from the correlator of a GPS receiver with INS data. An unscented Kalman filter (UKF), which employs a set of sigma points by deterministic sampling, avoids the error caused by linearization as in an extended Kalman filter (EKF). Based on the filter structural adaptation for describing various dynamic behaviours, the IMM nonlinear filtering provides an alternative for designing the adaptive filter in the ultra-tight GPS/INS integration. The use of IMM enables tuning of an appropriate value for the process of noise covariance so as to maintain good estimation accuracy and tracking capability. Two examples are provided to illustrate the effectiveness of the design and demonstrate the effective improvement in navigation estimation accuracy. A performance comparison among various filtering methods for ultra-tight integration of GPS and INS is also presented. The IMM based nonlinear filtering approach demonstrates the effectiveness of the algorithm for improved positioning performance.

Parallelized sigma-point Kalman filtering for structural dynamics

In this paper, the sigma-point Kalman filter (S-PKF) is adopted to track the state of composite structures undergoing impact-induced delamination. Estimates provided by the S-PKF are obtained through a set of sigma-points, which independently evolve in time according to the system dynamics. Since the number of sigma-points grows proportionally to the number of degrees of freedom of the space-discretized structural system, the S-PKF can become computationally demanding. Starting from the aforementioned independent evolution of the sigma-points, we propose a parallel implementation of the S-PKF within a shared-memory (OpenMP) architecture. Scalability and accuracy issues are eventually discussed.

Mobile Tracking in Mixed Line-of-Sight/Non-Line-of-Sight Conditions: Algorithm and Theoretical Lower Bound

The paper investigates the problem of mobile tracking in mixed line-of-sight (LOS)/non-line-of-sight (NLOS) conditions. The motion of mobile station is modeled by a dynamic white noise acceleration model, while the measurements are time of arrival (TOA). A first-order Markov model is employed to describe the dynamic transition of LOS/NLOS conditions. An improved Rao-Blackwellized particle filter (RBPF) is proposed, in which the LOS/NLOS sight conditions are estimated by particle filtering using the optimal trial distribution, and the mobile state is computed by applying approximated analytical methods. The theoretical error lower bound is further studied in the described problem. A new method is presented to compute the posterior Cramer-Rao lower bound (CRLB): the mobile state is first estimated by decentralized extended Kalman filter (EKF) method, then sigma point set and unscented transformation are applied to calculate Fisher information matrix (FIM). Simulation results show that the improved RBPF is more accurate than current methods, and its performance approaches to the theoretical bound.

Fuzzy Adaptive Interacting Multiple Model Nonlinear Filter for Integrated Navigation Sensor Fusion

In this paper, the application of the fuzzy interacting multiple model unscented Kalman filter (FUZZY-IMMUKF) approach to integrated navigation processing for the maneuvering vehicle is presented. The unscented Kalman filter (UKF) employs a set of sigma points through deterministic sampling, such that a linearization process is not necessary, and therefore the errors caused by linearization as in the traditional extended Kalman filter (EKF) can be avoided. The nonlinear filters naturally suffer, to some extent, the same problem as the EKF for which the uncertainty of the process noise and measurement noise will degrade the performance. As a structural adaptation (model switching) mechanism, the interacting multiple model (IMM), which describes a set of switching models, can be utilized for determining the adequate value of process noise covariance. The fuzzy logic adaptive system (FLAS) is employed to determine the lower and upper bounds of the system noise through the fuzzy inference system (FIS). The resulting sensor fusion strategy can efficiently deal with the nonlinear problem for the vehicle navigation. The proposed FUZZY-IMMUKF algorithm shows remarkable improvement in the navigation estimation accuracy as compared to the relatively conventional approaches such as the UKF and IMMUKF.

Non-symmetric unscented transformation with application to in-flight alignment

To overcome the extended Kalman Filter (EKF) limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformation. However, the ordinary UT has both local and global sampling problems. In this paper, non-symmetric unscented transformation (NSUT) is proposed to solve these problems. The sigma point sets of the NSUT are non-symmetrically located with respect to the zeroth sigma point set which is equal to the prior mean. Then, the weights of the sigma points are derived by using some conditions for mean and covariance. The non-symmetric unscented Kalman filter (NSUKF) based on the sigma points and weights of the NSUT are applied to the in-flight alignment (IFA) with a large heading error. With a low cost MEMS IMU, the simulations are carried out at an S-type trajectory to analyze the performance of the proposed method.

Unscented Kalman filtering in the additive noise case

The unscented Kalman filter (UKF) has four implementations in the additive noise case, according to whether the state is augmented with noise vectors and whether a new set of sigma points is redrawn from the predicted state (which is so-called resampling) for the observation prediction. This paper concerns the differences of performances for those implementations, such as accuracy, adaptability, computational complexity, etc. The conditionally equivalent relationships between the augmented and non-augmented unscented transforms (UTs) are proved for several sampling strategies that are commonly used. Then, we find that the augmented and non-augmented UKFs have the same filter results with the additive measurement noise, but only have the same state predictions with the additive process noise. Resampling is not believed to be necessary in some researches. However, we find out that resampling can be helpful for an adaptive Kalman gain. This will improve the convergence and accuracy of the filter when the large scale state modeling bias or unknown maneuvers occur. Finally, some universal designing principles for a practical UKF are given as follows: 1) for the additive observation noise case, it’s better to use the non-augmented UKF; 2) for the additive process noise case, when the small state modeling bias or maneuvers are involved, the non-resampling algorithms with state whether augmented or not are candidates for filters; 3) the resampling and non-augmented algorithm is the only choice while the large state modeling bias or maneuvers are latent.

Navigation Integration Using the Fuzzy Strong Tracking Unscented Kalman Filter

A navigation integration processing scheme, called the strong tracking unscented Kalman filter (STUKF), is based on the combination of an unscented Kalman filter (UKF) and a strong tracking filter (STF). The UKF employs a set of sigma points by deterministic sampling, such that the linearization process is not necessary, and therefore the error caused by linearization as in the traditional extended Kalman filter (EKF) can be avoided. As a type of adaptive filter, the STF is essentially a nonlinear smoother algorithm that employs suboptimal multiple fading factors, in which the softening factors are involved. In order to resolve the shortcoming in traditional approach for selecting the softening factor through personal experience or computer simulation, a novel scheme called the fuzzy strong tracking unscented Kalman filter (FSTUKF) is presented where the Fuzzy Logic Adaptive System (FLAS) is incorporated for determining the softening factor. The proposed FSTUKF algorithm shows promising results in estimation accuracy when applied to the integrated navigation system design, as compared to the EKF, UKF and STUKF approaches.