Unscented Transformation Research Articles

Closed-Form Algebraic Solutions for Angle-of-Arrival Source Localization With Bayesian Priors

This paper considers the problem of angle-of-arrival (AOA) source localization with Bayesian priors. Extending the so-called pseudolinear estimator (PLE) for AOA localization to the Bayesian framework results in the Bayesian PLE (BPLE). Similar to the PLE, the BPLE suffers from severe bias problems despite enjoying the inherent stability and computational simplicity of a closed-form least-squares estimator. Bayesian priors make bias compensation for the BPLE much more challenging than the PLE. We present a bias analysis for the BPLE from which two direct bias compensation methods are developed. As an alternative to direct bias compensation, a new approach with superior estimation performance and low computational complexity is then proposed by exploiting jointly the use of instrumental variables and 2-D unscented (or cubature) transformation. The resulting estimator, referred to as the bias-compensated Bayesian weighted instrumental variable estimator, enjoys the desirable properties of negligible bias and mean-squared error on par with the computationally demanding an iterative maximum a posteriori estimator. The comparative simulation studies are presented to corroborate the performance advantages of the proposed estimators.

Probabilistic Model for Microgrids Optimal Energy Management Considering AC Network Constraints

A new probabilistic approach for microgrids (MGs) optimal energy management considering ac network constraints is proposed in this paper. The economic model of an energy storage system (ESS) is considered in the problem. The reduced unscented transformation (RUT) is applied in order to deal with the uncertainties related to the forecasted values of load demand, market price, and available outputs of renewable energy sources (RESs). Moreover, the correlation between market price and load demand is taken into account. Besides, the impact of the correlated wind turbines (WT) on MGs’ energy management is studied. An enhanced JAYA (EJAYA) algorithm is suggested to achieve the best solution of the considered problem. The effective performance of the presented approach is verified by applying the suggested strategy on a modified IEEE 33-bus system. It can be observed that for dealing with probabilistic problems, the suggested RUT-EJAYA shows accurate results such as those of Monte Carlo (MC) while the computational burden (time and complexity) is lower.

Protocol-Based Unscented Kalman Filtering in the Presence of Stochastic Uncertainties

In this paper, the unscented Kalman filtering (UKF) problem is investigated for a class of general nonlinear systems with stochastic uncertainties under communication protocols. A modified unscented transformation is put forward to account for stochastic uncertainties caused by modeling errors. For preventing data collisions and mitigating communication burden, the round-robin protocol and the weighted try-once-discard protocol are, respectively, introduced to regulate the data transmission order from sensors to the filter. Then, by employing two kinds of data-holding strategies (i.e., zero-order holder and zero input) for those nodes without transmission privilege, two novel protocol-based measurement models are formulated. Subsequently, by resorting to the sigma point approximation method, two resource-saving UKF algorithms are developed, where the impact from the underlying protocols on the filter design is explicitly quantified. Finally, compared with the protocol-based extended Kalman filtering algorithms, a simulation example is presented to demonstrate the effectiveness of the proposed protocol-based UKF algorithms.

Trajectory prediction of cylinders freely dropped into water in two dimensions based on state space model

Experiments show that the behavior of dropped cylindrical objects may be stochastic if small disturbances exist at the initial moment. An envelope of various trajectories can be obtained from experimental tests. In addition, numerical results also indicate that small changes of initial states, e.g., drop angle, rolling frequency, and the selection of drag coefficient, affect the trajectory greatly. If the initial state follows the Gaussian distribution,it can be considered as a nonlinear prediction problem. Then the Unscented Transformation (UT) method, as a deterministic sampling method, can be used to predict the trajectory. In this paper, we firstly present a systematic formulation of the cylinder's three-degree-of-freedom (3DOF) of motions as a state space model. Secondly, the UT method and the interval estimation method are employed to solve this prediction problem. Finally, the predicted trajectories will be compared with the experimental data. Meanwhile, two performance parameters are first proposed to evaluate the accuracy of predicted trajectory. Both parameters can be directly applied to the salvage and recovery of dropped objects in the offshore operations. In addition, the numerical results also show that the UT method is an efficient way to predict the trajectory of cylinders freely dropped into water. • The state space model of a dropped cylinder has been successfully constructed in this paper. • The trajectory of a dropped cylinder is considered as the stochastic process due to the uncertainty at the initial moment. • The Unscented Transformation (UT) method is employed to solve the nonlinear trajectory prediction. • Two performance measures R A and R L have been first proposed to evaluate the accuracy of numerical trajectory prediction, in order to meet the requirement from the offshore engineering industry.

Stochastic optimization of cost-risk for integrated energy system considering wind and solar power correlated

Due to the growing penetration of renewable energies (REs) in integrated energy system (IES), it is imperative to assess and reduce the negative impacts caused by the uncertain REs. In this paper, an unscented transformation-based mean-standard (UT-MS) deviation model is proposed for the stochastic optimization of cost-risk for IES operation considering wind and solar power correlated. The unscented transformation (UT) sampling method is adopted to characterize the uncertainties of wind and solar power considering the correlated relationship between them. Based on the UT, a mean-standard (MS) deviation model is formulated to depict the trade-off between the cost and risk of stochastic optimization for the IES optimal operation problem. Then the UT-MS model is tackled by a multi-objective group search optimizer with adaptive covariance and Lévy flights embedded with a multiple constraints handling technique (MGSO-ACL-CHT) to ensure the feasibility of Perato-optimal solutions. Furthermore, a decision making method, improve entropy weight (IEW), is developed to select a final operation point from the set of Perato-optimal solutions. In order to verify the feasibility and efficiency of the proposed UT-MS model in dealing with the uncertainties of correlative wind and solar power, simulation studies are conducted on a test IES. Simulation results show that the UT-MS model is capable of handling the uncertainties of correlative wind and solar power within much less samples and less computational burden. Moreover, the MGSO-ACL-CHT and IEW are also demonstrated to be effective in solving the multi-objective UT-MS model of the IES optimal operation problem.

Parameter Identification and State of Charge Estimation of NMC Cells Based on Improved Ant Lion Optimizer

For lithium battery, which is widely utilized as energy storage system in electric vehicles (EVs), accurate estimating of the battery parameters and state of charge (SOC) has a significant effect on the prediction of energy power, the estimation of remaining mileage, and the extension of usage life. This paper develops an improved ant lion optimizer (IALO) which introduces the chaotic mapping theory into the initialization and random walk processes to improve the population homogeneity and ergodicity. After the elite (best) individual is obtained, the individual mutant operator is conducted on the elite individual to further exploit the area around elite and avoid local optimum. Then the battery model parameters are optimized by IALO algorithm. As for the SOC estimation, unscented Kalman filter (UKF) is a common algorithm for SOC estimation. However, a disadvantage of UKF is that the noise information is always unknown, and it is usually tuned manually by “trial-and-error” method which is irregular and time-consuming. In this paper, noise information is optimized by IALO algorithm. The singular value decomposition (SVD) which is utilized in the process of unscented transformation to solve the problem of the covariance matrix may lose positive definiteness. The experiment results verify that the developed IALO algorithm has superior performance of battery model parameters estimation. After the noise information is optimized by IALO, the UKF can estimate the SOC accurately and the maximum errors rate is less than 1%.

Quadrature-Based Nonlinear Joint Probabilistic Data Association Filter

This paper deals with the coupled problem of data association and nonlinear estimation for enhanced Space Situational Awareness (SSA) applications. High-order quadrature methods are used to implement a joint probabilistic data association (JPDA) filter to associate multiple measurements to multiple targets at any given point of time. It is shown that the use of higher-order quadrature methods achieves greater accuracy and stability. In particular, the recently developed conjugate unscented transformation is used to estimate the association probabilities and state estimates. Numerical simulations are used to illustrate the performance of the JPDA filter for SSA applications.

Simulation Performance Enhancement in Automotive Embedded Control Using the Unscented Transform

Automotive embedded systems comprise several domains, such as in software, electrical, electronics, and control. When designing and testing functions at the top level, one generally ignores the uncertainties arising from the electrical and electronic effects, which could lead to an irregular behavior and deteriorate their performance even using the appropriate methodology for designing the embedded control systems. Then, the studies and comparison on the effect of uncertainty in the automotive domain are important to improve the overall performance of those control systems. Here, we explored the uncertainty in control systems using the Monte Carlo (MC) and unscented transform (UT) methods. These methods have been applied to a mobile seat platform (MSP) and a light emitting diode (LED) used for lighting of heavy-duty vehicles. The UT for embedded control systems has shown better performance when compared to the Monte Carlo method, in order to reduce the number of required variables and computational resources in the simulation of failures and test-case generation. Finally, this investigation brings another application for the UT, in order to exemplify its applicability and advantages when compared with the other methods.

Stochastic Cascading Failure Model With Uncertain Generation Using Unscented Transform

This paper proposes a novel stochastic cascading failure (CF) model with full ac power flow using the unscented transform (UT) for power girds vulnerability analysis. The quasi-steady state (QSS) model is built upon previous dc power flow model by incorporating several enhancements. Our analysis shows that dc power flow does not provide an accurate estimation of the flow process under highly variable generation thus may underestimate the severity of the cascades. The incorporation of full ac power flow constraints allows us to access voltage profiles dynamics during CF in order to simulate voltage-related failures in the grid. The under-voltage load shedding relays are modeled along with a stochastic time-inverse overload relay to accurately simulate the protective system response. In addition, more realistic assumptions are considered in the modeling of wind power penetration using geographical information of grid topology and wind potential map for a given geographical area. The accuracy of the estimated flow process based on UT method is examined under different operating conditions in a 500-bus synthetic network. The proposed model is benchmarked against the historical blackout data and widely used models in the literature showing similar statistical patterns of blackout size.

A Quaternion-Based Robust Adaptive Spherical Simplex Unscented Particle Filter for MINS/VNS/GNS Integrated Navigation System

An improved filtering algorithm-robust adaptive spherical simplex unscented particle filter (RASSUPF) is proposed to achieve high accuracy, induce the amount of computation, and resist the influence of abnormal interference for the MINS/VNS/GNS integrated navigation system. This algorithm adopts spherical simplex unscented transformation (SSUT) to approximate the probability distribution, employs the spherical simplex unscented Kalman filter (SSUKF) to generate the importance sampling density of particle filter, and applies robust and adaptive estimation to control the influence of the abnormal information on the state model and the observation model. Simulation results demonstrate the proposed algorithm can effectively reduce the navigation error, improve the navigation positioning precision, and decrease the computation cost.

Event-Trigger Heterogeneous Nonlinear Filter for Wide-Area Measurement Systems in Power Grid

The development of distributed generation raises high requirement on accurate real-time state estimation with phasor measurement unit (PMU) for wide-area measurement systems (WAMSs). Although the particle filter is among the best in estimation performance, its computation burden is truly heavy toward smart sensors of current generation. Besides, the infrastructure for PMU data communication is confronted with the challenges of the amount of data boosted by the high sampling frequency of PMUs and booming grid size. This paper proposes an event-trigger heterogeneous nonlinear filter (ET-HNF) to solve these problems. A master–slave nonlinear filtering structure is designed for WAMS by taking full advantage of computation power both at generator node and estimation center. The local slave filter is established at generator node to perform the event-trigger strategy based on low computational unscented transformation. Moreover, a Monte Carlo-based filter served as the master, which improves estimation center’s filtering performance by cooperating with the center slave. Simulation results verify the feasibility and performance of ET-HNF using the standard IEEE 39-bus system with PMUs.

Unscented Kalman Filters for Riemannian State-Space Systems

Unscented Kalman Filters (UKFs) have become popular in the research community. Most UKFs work only with Euclidean systems, but in many scenarios it is advantageous to consider systems with state-variables taking values on Riemannian manifolds. However, we can still find some gaps in the literature's theory of UKFs for Riemannian systems: for instance, the literature has not yet i) developed Riemannian extensions of some fundamental concepts of the UKF theory (e.g., extensions of $\sigma$-representation, Unscented Transformation, Additive UKF, Augmented UKF, additive-noise system), ii) proofs of some steps in their UKFs for Riemannian systems (e.g., proof of sigma points parameterization by vectors, state correction equations, noise statistics inclusion), and iii) relations between their UKFs for Riemannian systems. In this work, we attempt to develop a theory capable of filling these gaps. Among other results, we propose Riemannian extensions of the main concepts in the UKF theory (including closed forms), justify all steps of the proposed UKFs, and provide a framework able to relate UKFs for particular manifolds among themselves and with UKFs for Euclidean spaces. Compared with UKFs for Riemannian manifolds of the literature, the proposed filters are more consistent, formally-principled, and general. An example of satellite attitude tracking illustrates the proposed theory.

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

AbstractThis study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.

Методика определения метрологических характеристик зондовой установки для измерения параметров рассеяния изделий СВЧ наноэлектроники на подложке

The method for determining the metrological characteristics of the probe system for the scattering parameter measurements of passive and active devices on a semiconductor substrate is presented. The main idea is to estimate the residual errors and determine the accuracy of the obtained estimates. The implementation of the idea is based on a special filtration technology by the processing of verification measurements and unscented transformation for the study of the quality of a verification set which consists of a single transmission line that can be made on a substrate with devices.

A protection scheme for microgrids using time-time matrix z-score vector

This paper presents a new scheme for microgrid protection using the Time-Time (TT) transform. Initially, the bus currents are processed through the TT-transform to compute the TT-matrixes. To determine the fault patterns, an appropriate index based on the TT-matrix z-score vector is introduced. A threshold is used for fault detection and classification within a few cycles from the fault inception for both grid-connected and islanded modes. A threshold selection approach using the unscented transformation (UT) is used to accommodate low signal-to-noise ratios (SNRs). The protection scheme is evaluated through extensive simulations using PSCAD/EMTDC software for two different microgrids (MGs). Various test scenarios consisting of all shunt and high impedance faults are investigated in MGs with radial and loop topologies for grid-connected and autonomous modes of operation. The results verify that the proposed scheme can reliably detect and classify faults. Furthermore, the proposed method is robust to changes in fault parameters, and performs well under highly noisy conditions.

Unscented Kalman Filtering for Nonlinear Systems with Colored Measurement Noises and One-Step Randomly Delayed Measurements

Traditional unscented Kalman filtering (UKF) cannot solve the filtering problem for nonlinear systems with colored measurement noises and one-step randomly delayed measurements. To fix this problem, a new UKF algorithm is proposed in this paper. First, a system model with one-step randomly delayed measurements and colored measurement noises is established, wherein a first order Markov sequence model for whitening colored noises and an independently identical distributed Bernoulli variable for modeling one-step randomly delayed measurements is introduced. Second, an UKF is proposed for the above established models through unscented transformation by calculating the nonlinear states posterior mean and covariance based on the Bayesian filter framework. Specially, the proportional symmetric sampling method is used in the new UKF algorithm. Finally, the effectiveness and superiority of the proposed method is verified via simulation.

Artificial Neural Networks with Random Weights for Incomplete Datasets

In this paper, we propose a method to design Neural Networks with Random Weights in the presence of incomplete data. We present a method, under the general assumption that the data is missing-at-random, to estimate the weights of the output layer as a function of the uncertainty of the missing data estimates. The proposed method uses the Unscented Transform to approximate the expected values and the variances of the training examples after the hidden layer. We model the input data as a Gaussian Mixture Model with parameters estimated via a maximum likelihood approach. The validity of the proposed method is empirically assessed under a range of conditions on simulated and real problems. We conduct numerical experiments to compare the performance of the proposed method to the performance of popular, parametric and non-parametric, imputation methods. By the results observed in the experiments, we conclude that our proposed method consistently outperforms its counterparts.

Unscented Transformation-Based Multi-Robot Collaborative Self-Localization and Distributed Target Tracking

The problem of multi-robot collaborative self-localization and distributed target tracking in practical scenarios is studied in this work. The major challenge in solving the problem in a distributed fashion is properly dealing with inter-robot and robot–target correlations in order to realize consistent state estimates of the local robots and the target simultaneously. In this paper, an unscented transformation-based collaborative self-localization and target tracking algorithm is proposed. Inter-robot correlations are approximated in a distributed fashion, and robot–target correlations are safely discarded with a conservative covariance intersection method. Furthermore, the state update is realized in an asynchronous manner with different kinds of measurements while accounting for measurement and communication limitations. Finally, to deal with nonlinearity in the processes and measurement models, the unscented transformation approach is adopted. Unscented transformation is better able to characterize nonlinearity than the extended Kalman filter-based method and does not require computation of the Jacobian matrix. Simulations are extensively studied to show that the proposed method can realize stable state estimates of both local robots and targets, and results show that it outperforms the EKF-based method. Moreover, the effectiveness of the proposed method is verified on experimental quadrotor platforms carrying off-the-shelf onboard sensors.

A robust unscented transformation for uncertain moments

This paper proposes a robust version of the unscented transform (UT) for one-dimensional random variables. It is assumed that the moments are not exactly known, but are known to lie in intervals. In this scenario, the moment matching equations are reformulated as a system of polynomial equations and inequalities, and it is proposed to use the Chebychev center of the solution set as a robust UT. This method yields a parametrized polynomial optimization problem, which in spite of being NP-Hard, can be relaxed by some algorithms that are proposed in this paper.

A novel intelligent SOC prediction method of lithium-ion battery packs based on the improved unscented transformation

The development of power lithium-ion battery packs is limited by the research and development level of Battery Management System (BMS). In this paper, an improved unscented transformation method is proposed, which is based on the six-section aero-lithium battery pack as the object of detection. The effective iterative calculation of State of Charge (SOC) value is realized by simplifying the process of three-particle and double Sigma processing. The accuracy of the method is verified by comparing the simulation results with the actual measurement results. Experimental results show that the error of this method is less than 3.00%, which effectively improves the SOC estimation accuracy and is of great significance to energy management and safety assurance of power lithium-ion battery packs.