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This letter presents explicit sub-optimal track-to-track fusion algorithms for Multi-Sensor Systems (MSS) estimating nonlinear processes. The individual tracks in an MSS are correlated due to the presence of a common process noise in the track estimation errors. Herein, we propose recursive formulae for consistent correlation estimation in mildly and highly nonlinear systems that respectively use Extended Kalman Filters (EKF) and Unscented Kalman Filters (UKF) for track estimation. In a mildly nonlinear system, the linearized model employed in the EKF-based MSS architecture offers a correlation propagation formula whose coupling with the optimal track fusion rule generates a sub-optimal fused estimate. On the other hand in highly nonlinear systems, the UKF-based architectures are proven effective for track estimation. The UKF works based on the unscented transform of deterministic sigma points, which is equivalent to the Statistical Linearization Regression (SLR) process. For UKF-based MSS architectures, we propose a consistent correlation propagation recursion according to the SLR technique that will be coupled with the optimal track fusion rule to generate a sub-optimal fused estimate. The performance of the developed fusion algorithms is demonstrated through conducting a statistical test and an average root mean square error analysis.

In this paper, we propose a novel computer vision system for tracking high-speed non-rigid skaters over a large playing area in short track speed skating competitions. Several important features distinguish the proposed approach from others: 1. Introducing the reference frames as a transition through which each frame can be mapped to the field model to reduce the error accumulation of the projection, and it's very important for a long video sequence and helpful for improving the precision of the system. 2. Incorporating the hierarchical model based on the contextual knowledge and multiple cues into the unscented Kalman filter to improve the tracking performance when occlusions occur. 3. Choosing the unscented Kalman filter for visual tracking in the sports domain, it is superior to EKF in theory, and is more efficient than particle filter. 4. Evaluating the relation between the accuracy of homography H t -1,t and tracking performance. Proposing a novel and objective evaluation method to measure the precision of our practical system. However, The main problem is remained in our current system: how to improve the tracking performance when skaters are moving in groups during a long and continuously full occlusion. In future, we can model the skater's uniform of different teams in each sub-region, and the uniform model can be used to assist in tracking the target occluded for a long time. 5.

Active estimation is becoming a more important issue in control theory and its application, especially in the nonlinear control of uncertain systems, such as robots and unmanned vehicles where time-varying parameters and uncertainties exist extensively in the dynamics and working environment. Among the available techniques for active modeling, Neural Networks (NN) and NN-based self learning have been proposed as one of the most effective approaches in 1990s (Pesonen et al., 2004). However the problems involved in NN, such as training data selection, online guaranteed convergence, robustness, reliability and real-time implementation, still remain open and limit its application in real systems, especially those requiring high reliable control. Most recently, the encouraging achievements in sequential estimation makes it becoming an important direction for online modeling and model-reference control (Napolitano, et al., 2000). Among stochastic estimations, the most popular one for nonlinear system is the Extended Kalman Filter (EKF). Although widely used, EKF suffers from the deficiencies including the requirement of sufficient differentiability of the state dynamics, the susceptibility to bias and divergence during the estimation. Unscented Kalman Filter (UKF) (Julier et al., 1995; Wan & Van der Merwe, 2000) provides a derivative-free way to the state parameter estimation of nonlinear systems by introducing the so called ‘unscented transformation’, while achieving the second-order accuracy (the accuracy of EKF is first order) with the same computational complexity as that of EKF. Although the nonlinear state dynamics are used without linearization and the calculations on Jacobians or Hessians are not involved, UKF still falls into the framework of Kalman-type filters, which can only achieve good performance under a priori assumptions (Jazwinski, 1970), which includes: 1) accurate reference models, 2) complete information of the noise distribution, and 3) proper initial conditions. However, such a priori knowledge is often not accurate, or even not available in practice. The normal UKF will suffer from performance degradation or even instability due to the mismatch between the a priori assumptions and the real ones within the system to be controlled. One of the approaches solving this problem is to introduce adaptive mechanism into a normal filter, i.e., the adaptive law automatically tunes the filter parameters to match the O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg

The extended Kalman filter (EKF) is one of the most widely used methods for state estimation with communication and aerospace applications based on its apparent simplicity and tractability (Shi et al., 2002; Bolognani et al., 2003; Wu et al., 2004). However, for an EKF to guarantee satisfactory performance, the system model should be known exactly. Unknown external disturbances may result in the inaccuracy of the state estimate, even cause divergence. This difficulty has been recognized in the literature (Reif & Unbehauen, 1999; Reif et al., 2000), and several schemes have been developed to overcome it. A traditional approach to improve the performance of the filter is the 'covariance setting' technique, where a positive definite estimation error covariance matrix is chosen by the filter designer (Einicke et al., 2003; Bolognani et al., 2003). As it is difficult to manually tune the covariance matrix for dynamic system, adaptive extended Kalman filter (AEKF) approaches for online estimation of the covariance matrix have been adopted (Kim & ILTIS, 2004; Yu et al., 2005; Ahn & Won, 2006). However, only in some special cases, the optimal estimation of the covariance matrix can be obtained. And inaccurate approximation of the covariance matrix may blur the state estimate. Recently, the robust H∞ filter has received considerable attention (Theodor et al., 1994; Shen & Deng, 1999; Zhang et al., 2005; Tseng & Chen, 2001). The robust filters take different forms depending on what kind of disturbances are accounted for, while the general performance criterion of the filters is to guarantee a bounded energy gain from the worst possible disturbance to the estimation error. Although the robust extended Kalman filter (REKF) has been deeply investigated (Einicke & White, 1999; Reif et al., 1999; Seo et al., 2006), how to prescribe the level of disturbances attenuation is still an open problem. In general, the selection of the attenuation level can be seen as a tradeoff between the optimality and the robustness. In other words, the robustness of the REKF is obtained at the expense of optimality. This chapter reviews the adaptive robust extended Kalman filter (AREKF), an effective algorithm which will remain stable in the presence of unknown disturbances, and yield accurate estimates in the absence of disturbances (Xiong et al., 2008). The key idea of the AREKF is to design the estimator based on the stability analysis, and determine whether the error covariance matrix should be reset according to the magnitude of the innovation. O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg

In this paper we develop the Multiple Model Particle Filter (MMPF) for nonlinear systems. The particle filter is used to estimate the conditional probability for the modes while the Conjugate Unscented Transform (CUT) based Kalman filter is used to estimate the dynamical state of the system. The resultant particle filter is a bank of nonlinear filters with the sequence of modes given by the samples. A tracking example is used to illustrate the working and efficacy of the MMPF with respect to the Interactive Multiple Model (IMM) filter. Comparison results are shown using the Extended Kalman filter and Conjugate Unscented Kalman filter for both the MMPF and IMM filters.

Manipulating unmanned aerial vehicles (MUAVs) are aerial robots equipped with a mechanism to physically interact with the environment. State estimation of such robots is a challenging problem due to inherent couplings, nonlinearities and uncertainties of MUAV complex dynamics and, therefore, popular algorithms such as extended Kalman filter may not be applicable. With the above considerations, this paper formulates two variants of the unscented Kalman filter using (i) general and (ii) spherical unscented transform to address state estimation problem in MUAVs. In order to examine the effect of estimation quality on overall control performance, first the coupled dynamics of a quadcopter endowed with a robotic manipulator is presented. Next, a linear–quadratic–Gaussian (LQG) control is designed to achieve simultaneous control of the quadcopter and its manipulator. Then, the performance of each unscented Kalman filter algorithm is compared with that of extended Kalman filter in the context of estimation accuracy, overall control performance, and algorithm execution time. Additionally, sensitivity of the proposed approaches to increasing noise levels and total loss of sensory data are examined.

The time and measurement update for the discrete time Kalman filter can be formulated in terms of conditional means and covariances. The unscented Kalman filter can be interpreted as calculating these conditional means and covariances by using the unscented transform. This approach can also be directly applied to nonlinear models as an alternative to the discrete time extended Kalman filter. In this paper, a novel method for computing the unscented Kalman filter for a nonlinear model with continuous time dynamics and discrete time measurements is presented. Compared to the existing approaches, this method is far simpler and less computationally demanding, and it performs at least as well.

Due to the communication channel noises, GPS synchronization process, changing environment temperature and different operating conditions of the system, the statistics of the system process and measurement noises may be unknown and they may not follow Gaussian distributions. As a result, the traditional Kalman filter-based dynamic state estimators may provide strongly biased state estimates. To address these issues, this paper develops a robust generalized maximum-likelihood unscented Kalman filter (GM-UKF). The statistical linearization approach is presented to derive a compact batch-mode regression form by processing the predicted state vector and the received measurements simultaneously. This regression form enhances the data redundancy and allows us to detect bad phasor measurement unit measurements and incorrect state predictions, and filter out unknown Gaussian and non-Gaussian noises through the generalized maximum likelihood-estimator. The latter minimizes a convex Huber function with weights calculated via the projection statistics (PS). Particularly, the PS is applied to a proposed 2-dimensional matrix that consists of temporally correlated innovation vectors and predicted states. Finally, the total influence function is used to derive the error covariance matrix of the GM-UKF state estimates, yielding the robust state prediction at the next time instant. Extensive simulations carried out on the IEEE 39-bus test system demonstrate the effectiveness and robustness of the proposed method.

In this paper, we present robust localization algorithms that use range measurements. The least median of squares (LMedS)-weighted least squares (WLS), LMedS-spherical simplex unscented transform (SSUT) based WLS and Tukey-based extended Kalman filter (EKF) algorithms are proposed for line-of-sight (LOS)/non-line-of-sight (NLOS) mixture environments. First, the LMedS solution is obtained, and then sensors are predicted to be LOS or LOS/NLOS mixture sensors. The range observation predicted as an outlier is replaced with the estimated distance obtained using the LMedS algorithm. Subsequently, the two-step WLS method is executed using these new distance measurements. In the Tukey-based EKF method, Tukey's risk function and the 3-σ edit rule are employed in the innovation step. Furthermore, the mean square error (MSE) analysis of the proposed algorithms is performed. We demonstrate that the positioning accuracy of the proposed methods is higher than that of conventional methods through extensive simulation.

This paper explores multiple model adaptive estimation &#x0028 MMAE &#x0029 method, and with it, proposes a novel filtering algorithm. The proposed algorithm is an improved Kalman filter — multiple model adaptive estimation unscented Kalman filter &#x0028 MMAE-UKF &#x0029 rather than conventional Kalman filter methods, like the extended Kalman filter &#x0028 EKF &#x0029 and the unscented Kalman filter &#x0028 UKF &#x0029. UKF is used as a subfilter to obtain the system state estimate in the MMAE method. Single model filter has poor adaptability with uncertain or unknown system parameters, which the improved filtering method can overcome. Meanwhile, this algorithm is used for integrated navigation system of strapdown inertial navigation system &#x0028 SINS &#x0029 and celestial navigation system &#x0028 CNS &#x0029 by a ballistic missile’s motion. The simulation results indicate that the proposed filtering algorithm has better navigation precision, can achieve optimal estimation of system state, and can be more flexible at the cost of increased computational burden.

In this paper, an unscented Kalman filter (UKF) for a nonlinear fractional-order system (FOS) with the unknown order is proposed. The continuous-time nonlinear FOS is discretized by Grunwald-Letnikov difference, and the corresponding difference equation is obtained. Besides, the augmented vector is presented to establish the augmented equation with respect to the state and fractional-order, and the nonlinear functions in the nonlinear augmented equation are performed by the unscented transform. The state and unknown order are estimated by the proposed adaptive UKF to improve the accuracy of state and order estimations compare with the extended Kalman filter. Finally, two examples are given to verify the effectiveness.

International audience; We consider the problem of localization and navigation of Autonomous Underwater Vehicles (AUV) in the context of high performance subsea asset inspection missions in deep water. We propose a solution based on the recently introduced Unscented Kalman Filter on Manifolds (UKF-M) for onboard navigation to estimate the robot's location, attitude and velocity, using a precise round and rotating Earth navigation model. Our algorithm has the merit of seamlessly handling nonlinearity of attitude, and is far simpler to implement than the extended Kalman filter (EKF), which is widely used in the navigation industry. The unscented transform notably spares the user the computation of Jacobians and lends itself well to fast prototyping in the context of multi-sensor data fusion. Besides, we provide the community with feedback about implementation, and execution time is shown to be compatible with real-time. Realistic extensive Monte-Carlo simulations prove uncertainty is estimated with accuracy by the filter, and illustrate its convergence ability. Real experiments in the context of a 900m deep dive near Marseille (France) illustrate the relevance of the method.

Inertial Navigation Systems (INS) are the main part of the integrated navigation for most of the aerial vehicles. However, the accuracy of an inertial navigation solution decreases with time as the inertial instrument (e.g., gyroscope and accelerometer) errors are integrated through the navigation equations. Therefore, different aiding techniques are used to bound the drift in these systems. One of the commonly used techniques is the integration of INS with Global Navigation Satellite System (GNSS) signals. By means of this integration, the advantages of both technologies are combined to give a complete navigation solution. The need for Terrain Referenced Navigation (TRN) arises when these satellite based radio signals are unavailable. In recent years, research on the application of TRN to aerial vehicles has been increased rapidly with the developments in the accuracy of digital terrain elevation database (DTED). Since the land profile is inherently nonlinear, TRN becomes a nonlinear estimation problem. Because of the highly nonlinear problem, linear or linearized estimation techniques such as Kalman or Extended Kalman Filter (EKF) do not work properly for many terrain profiles. Hence, this paper focuses on nonlinear filtering techniques and presents the main principles of two different TRN methods. These methods will be compared and advantages of both methods will be presented. The first method is the Sequential Monte Carlo (SMC) technique namely the particle filter (PF) for dealing with nonlinearities and different types of probability distributions even multi-modal. PF is an approximate optimal filter on correct model and based on particle representation of probability density function. The second method is the Unscented Kalman Filter (UKF) based on the Unscented Transform (UT) of sigma points. The basic idea is to approximate the probability density function with deterministically selected and weighted small number of sigma points. Simulations with different inertial measurement units (IMUs), with different initial errors, over maps with various resolutions are performed and investigated. The performance of both nonlinear filtering algorithms will be presented through Monte Carlo simulations.

This paper revisits the problem of uncertainty propagation in robotic systems. Standard approaches use Kalman filtering, either Extended (linearized) (EKF) or Unscented Transform-based (UKF) Kalman Filters to handle state estimation of robotic systems. These dynamic filters operate on predict-correct cycle where the current state is predicted based on prior state, and then corrected based on current observations using calculated correction weight. These approaches provide both mean and uncertainty (covariance) of the state estimates. Alternative state estimation techniques include Monte-Carlo based particle filters. The system state estimate is often used in composition with other transforms to localize objects in a world frame. Often these internal transforms are treated as known nominal quantities, and their uncertainty is ignored. In this paper, we revisit the handling of errors and compare several representations to highlight limitations of current approaches.

The co-estimation of the ultracapacitor states and parameters using the Unscented Kalman Filter (UKF) is fulfilled in this paper. In the proposed method, the state-space model of the ultracapacitor based on a three-branch equivalent circuit model is used for joint state and parameter estimation. To demonstrate the observability of this model, the dynamic interdependence of the system states is analyzed using a graphical approach (GA). Unlike the KF and Extended KF (EKF) methods that use linearized system models, the UKF uses a nonlinear transformation (unscented transform), which improves the estimation accuracy. In addition, the correlations between the states and parameters are taken into account during the estimation process, which further improves the state estimation accuracy. Some experiments are conducted in the laboratory for testing the usefulness of the suggested approach. The results demonstrate that the suggested approach can provide accurate state and parameter estimation results while it is also able to run in real-time on low-cost digital signal processors (DSPs).

For continuous-discrete filtering with unpredictable approximation errors, by proposing the novel stochastic feedback scheme, this note elaborates a closed-loop adaptive Kalman filter for nonlinear continuous-discrete systems. In conventional filters, unknown approximation errors might arise due to the integration/discretization and linearization of continuous model, and ruin the optimality of Kalman theory. As the main contribution of this note, the stochastic feedback based covariance adaption scheme does not require the approximation steps; instead, the posteriori sequence is mined as a feedback to adapt the priori error covariance, so that the unpredictable errors and costly calculations can be reduced or controlled in the novel closed-loop filtering structure. The new approach's advantages in computational cost, adaptability, and accuracy have been demonstrated by the numerical simulations.

Electric vehicles (EVs) require reliable and very accurate battery state-of-charge (SOC) estimation to maximize their performance. A commonly used estimation technique, the extended Kalman filter (EKF), provides an accurate estimate of the SOC. However, EKF has some limitations, such as it assumes the knowledge of the statistics of the process noise and measurement noise is available, which practically cannot be guaranteed. In this paper, an adaptive equivalent-circuit model is proposed and used for SOC estimation. The proposed model is based on a common cell model with adaptive parameters tracking feature implemented using an artificial neural network controller embedded within the model. A variant of the EKF, namely the unscented Kalman filter (UKF), is used to achieve more accurate estimates of the SOC with a relatively fast convergence speed. The UKF uses the unscented transform to obtain the statistics of the process noise covariance. Furthermore, the autocovariance least-squares technique is used to estimate the measurement noise covariance by accounting for possible correlation in the measurement innovations, which enhances the accuracy of the estimate. Derivation of the proposed method followed by experimental verification is presented in this paper.

This work, based on the standard unscented Kalman filter (UKF), proposes a modified UKF for highly non-linear stochastic systems, assuming that the associated probability distributions are normal. In the standard UKF with 2n + 1 sample points, the estimated mean and covariance match the true mean and covariance up to the third order, besides, there exists a scaling parameter that plays a crucial role in minimising the fourth-order errors. The proposed method consists of a computationally efficient formulation of the unscented transform that incorporates N − 1 , N ≥ 2 , constant parameters to scale 2 n ( N − 1 ) + 1 sample points such that the 2 N th order errors are minimised. The scaling parameters are obtained by solving a set of algebraic equations. Through rigorous analytical processes and numerical simulations, it is demonstrated that the new filter provides consistent estimates and the estimation error of the modified UKF is smaller than that of the standard UKF. With the help of a well-studied case, univariate non-stationary growth model, the authors evaluate the estimation performance of the new technique using 4n + 1 sample points over 100 independent runs.

This paper presents a robust dynamic state estimator for the synchronous generators with unknown parameters. The estimator uses a constrained iterated unscented Kalman filter to estimate the state variables and unknown parameters of a two-axis model of a synchronous generator. The developed estimator's performance is validated using simulations, where the estimator is subjected to arbitrary initialization and large parameter errors. The developed dynamic estimator can potentially be used not only to track the dynamic states but also to detect and identify changes in model parameters with little a priori knowledge about the parameters other than a broad range which can be specified via appropriate constraints.

Laboratory experiments next to a variety of observations, especially in subduction zones, have explored the existence of a premonitory stable slow slip growth phase preceding large earthquakes. These phenomena play an important role in the earthquake cycle and thus precise imaging and monitoring of these events are of great significance. In the literature, ENIF (extended network inversion filter) has been proposed as a rigorous algorithm capable of isolating signal from different types of noise and thereby provides us with deep insight into spatio-temporal evolution of slow slip events. Despite its considerable advantages, ENIF still suffers from some limitations. ENIF applies Tikhonov method of regularization with a quadratic form of cost function. While anomalous slip regions have clear contrast with the background slip in reality, Tikhonov regularization tends to over smooth (globally smooth) the slipping portion on the estimated images. In order to avoid over smoothing phenomenon, we have incorporated into ENIF an image segmentation step which tries to preserve edges of slow-slip event. As a second limitation, due to the nonlinearity imposed by such constraint as non-negativity of slip rate, uncertainty propagation through model is not simple. As the core of ENIF, EKF (extended Kalman filter), performs uncertainty propagation by linearization of nonlinear model using Jacobian and Hessian matrices. As an alternative for EKF, we have also investigated the application of UKF (unscented Kalman filter) which uses UT (unscented transform) for uncertainty propagation. Finally, we tested our proposed algorithm using a low signal to noise ratio synthetic data set. The results show a significant improvement in the performance of ENIF when the segmentation step is incorporated into the algorithm. Keywords: Slow-slip earthquakes, Kalman filter, Over smoothing, Image segmentation, Unscented transform

Having an accurate mathematical model is essential in design, control, and navigation process of an autonomous underwater vehicle (AUV). Due to the modeling simplifications, the available mathematical models suffer from the uncertainty of their parameters and they usually need an identification phase for improving the modeling accuracy. In case of AUVs, the hydrodynamic coefficients of the model play an important role and they should be identified using the available experimental data. In this paper, a 6-DoF dynamic model of an AUV is presented and then some of its parameters including viscous damping, the body lift, and control input coefficients that have the highest effects on the modeling error are identified by augmented state model method. The extended Kalman filter (EKF), cubature Kalman filter (CKF), and transformed unscented Kalman filter (TUKF) are used as the estimation filters. To verify and compare the three estimation filters with consideration of the predetermined hydrodynamic coefficients and spiral maneuver AUV results, these three methods are evaluated. The results indicate that the TUKF identifies the best hydrodynamic model due to solving both the CKF nonlocal sampling problem and the EKF linearization problem.

The joint estimation of direction of arrivals (DOA) and carrier frequencies of band-limited source signals is considered in this paper. A novel technique based on nonlinear Kalman filters is proposed for this joint angular and spectral estimation problem for cognitive radio (CR). Since sampling a wideband spectrum at Nyquist rate increases the analog-to-digital converter (ADC) requirements, we propose executing Kalman filter algorithm over a spatial state space model. Thus, one time sample is required and hardware complexity is reduced. Two types of nonlinear Kalman filters, extended Kalman filter (EKF) and unscented Kalman filter (UKF), are proposed. We consider their sub-optimal performance and show how to control their convergence. However, the proposed algorithms can detect a number of source signals limited to the number of elements in employing arrays.

As the fault and unknown inputs may cause great estimate errors and even divergence for conventional filters when dealing with nonlinear discrete-time systems, this paper proposed a three-stage unscented Kalman filter (ThSUKF) and robust three-stage unscented Kalman filter (RThSUKF) to improve the performance by estimating the state and fault simultaneously. The detail derivation of the filters is illustrated in the paper. The proposed filters are available when the statistic models of the fault and unknown inputs are perfectly known. But if there is no prior knowledge about the fault and unknown input, RThSUKF can also have a great performance. Their performance and effectiveness are demonstrated by the numerical simulation of two-dimensional target tracking. And the results show that ThSUKF and RThSUKF can achieve superior performances with low estimation errors if the nonlinear systems are in presence of fault and unknown inputs.

This paper is concerned with the distributed filtering problem for discrete-time nonlinear systems over a sensor network. In contrast with the distributed filters with linear consensus estimate, a distributed extended Kalman filter (EKF) is developed with nonlinear consensus estimate. Specifically, a new nonlinear consensus protocol with polynomial form is proposed to generate the consensus estimate. By using the variance-constrained approach, the Kalman gain matrix is determined for each node to guarantee an optimized upper bound on the state estimation error covariance despite consensus terms and linearization errors. It is shown that the Kalman gain matrix can be derived by solving two Riccati-like difference equations. The effectiveness of the proposed filter is evaluated on an indoor localization of a mobile robot with visual tracking systems.

The objective of this study was to improve the performance of starlight refraction navigation using the unscented Kalman filter. In previous studies, the uncertainty and nonlinearity of the starlight refraction model limited the advancement of navigation accuracy. To overcome this limitation, an adaptive unscented Kalman filter is proposed in which the sampling weights are selected as the adaptive parameters. During the filtering process, the sampling weights are adaptively tuned based on the gradient descent method to increase the accuracy of the unscented transform predicted value. Simulations were conducted to verify the proposed filter. Results showed that it notably improved the navigation accuracy, and the position error was reduced to 68.7 m.

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

This work proposes constrained constant modulus unscented Kalman filter (CCM-UKF) algorithm and its low-complexity version called reduced-rank constrained constant modulus unscented Kalman filter (RR-CCM-UKF) algorithm for blind adaptive beamforming. In the generalized sidelobe canceller (GSC) structure, the proposed algorithms are devised according to the CCM criterion. Firstly, the cost function of the constrained optimization problem is transformed to suit the Kalman filter-style state space model. Then, the optimum weight vector of the beamformer can be estimated by using the recursive formulas of UKF. In addition, the a priori parameters of UKF (system and measurement noises) are processed adaptively in the implementation. Simulation results demonstrate that the proposed algorithms outperform the existing methods in terms of convergence speeds, output signal-tointerference- plus-noise ratios (SINRs), mean-square deviations (MSDs) and robustness against steering mismatch.

A robust unscented Kalman filter based on a multiplicative quaternion-error approach is proposed for nanosat estimation in the presence of measurement faults. The global attitude parameterization is given by a quaternion, while the local attitude error is defined using a generalized three-dimensional attitude representation. The proposed algorithm uses a statistical function including measurement residuals to detect measurement faults and then uses an adaptation scheme based on multiple measurement scale factor for filter robustness against faulty measurements. The proposed algorithm is demonstrated for the attitude estimation of a nanosat with an on-board three-axis magnetometer and rate-integrating gyros in the presence of measurement faults as well as satellite orbit errors. To compare the estimation performance of the proposed algorithm, the robust unscented Kalman filter with single measurement noise scale factor, the standard extended Kalman filter and the unscented Kalman filter are also implemented under the same simulation conditions.

Summary LiFePo4 battery is widely used in electric vehicles; however, its flatness and hysteresis of the open-circuit voltage curve pose a big challenge to precise state of charge (SOC) estimation. The issue is discussed and addressed in this paper. First, a cell model with hysteresis is built to describe real-time dynamic characteristics of the LiFePo4 battery. Second, the model parameters and SOC are estimated independently to avoid the possibility of cross interference between them. For model identification, an adaptive unscented Kalman filter (AUKF) algorithm is used to identify the cell parameters as they change slowly. While SOC could change rapidly, wavelet transform AUKF algorithm is put forward to estimate SOC. In the novel algorithm, the measurement noise can be estimated and updated online. Finally, the performance of the proposed method is verified under dynamic current condition. The experimental results show that estimated value based on the proposed method is more accurate than unscented Kalman filter-based method and AUKF-based algorithm. Meanwhile, the proposed estimator also has the merits of fast convergence and good robustness against the initialization uncertainty.

We propose a cascaded Kalman filtering (CKF) technique, a series of extended Kalman filter (EKF), and a linear Kalman filter, for joint tracking of phase and amplitude noise as well as polarization state. We further propose an adaptive CKF (A-CKF), in which the process noise covariance has been estimated adaptively, using covariance matching method. Numerical results on back-to-back and transmission scenarios on 28-Gbd polarization-multiplexed 16-quadrature amplitude modulation system justify that our proposed CKF as well as A-CKF outperforms the earlier proposed EKF and the unscented Kalman filter (UKF) with an improved tolerance toward polarization rotations and phase noise besides less computational effort.

An interlaced matrix Kalman filter, which is based on vector observations and gyro measurements, is proposed for spacecraft attitude estimation in this paper. It combines the matrix Kalman filter and cubature Kalman filter to estimate spacecraft attitude and gyro drift bias, respectively. The defects of the original matrix Kalman filter, which could only estimate the attitude parameters of spacecraft, are addressed by the proposed interlaced matrix Kalman filter. In addition, the dimensions of cubature Kalman filter for conventional attitude estimation method are reduced by the designed recursive algorithm. It is noted that the two filters are not independent with each other. Firstly, the attitude quaternion of spacecraft is estimated by the modified matrix Kalman filter. Then, the estimated quaternion is input for the recursive cubature Kalman filter, which is used to estimate the gyro drift bias. Finally, the estimated gyro drift bias is compensated for the measurements of the gyros. Therefore, the precision of the estimated attitude of spacecraft is improved by the interacting process of the modified matrix Kalman filter and recursive cubature Kalman filter. A simulation test is designed to verify the advantage of the proposed method by comparing with the previous method, and the results indicate that the proposed algorithm has better performance on convergence rate and stability.

Limited by the low precision of small satellite sensors, the estimation theories with high performance remains the most popular research topic for the attitude estimation. The Kalman filter (KF) and its extensions have been widely applied in the satellite attitude estimation and achieved plenty of achievements. However, most of the existing methods just take use of the current time-step’s priori measurement residuals to complete the measurement update and state estimation, which always ignores the extraction and utilization of the previous time-step’s posteriori measurement residuals. In addition, the uncertainty model errors always exist in the attitude dynamic system, which also put forward the higher performance requirements for the classical KF in attitude estimation problem. Therefore, the novel robust double gain unscented Kalman filter (RDG-UKF) is presented in this paper to satisfy the above requirements for the small satellite attitude estimation with the low precision sensors. It is assumed that the system state estimation errors can be exhibited in the measurement residual; therefore, the new method is to derive the second Kalman gain K k 2 for making full use of the previous time-step’s measurement residual to improve the utilization efficiency of the measurement data. Moreover, the sequence orthogonal principle and unscented transform (UT) strategy are introduced to robust and enhance the performance of the novel Kalman Filter in order to reduce the influence of existing uncertainty model errors. Numerical simulations show that the proposed RDG-UKF is more effective and robustness in dealing with the model errors and low precision sensors for the attitude estimation of small satellite by comparing with the classical unscented Kalman Filter (UKF).

Bearing-only passive target tracking is a well-known underwater defence issue dealt in the recent past with the conventional nonlinear estimators like extended Kalman filter (EKF) and unscented Kalman filter (UKF). It is being treated now-a-days with the derivatives of EKF, UKF and a highly sophisticated particle filter (PF). In this paper, two novel methods based on the Estimate Merge Technique are proposed. The Estimate Merge Technique involves a process of getting a final estimate by the fusion of a posteriori estimates given by different nonlinear estimates, which are in turn driven by the towed array bearing-only measurements. The fusion of the estimates is done with the weighted least squares estimator (WLSE). The two novel methods, one named as Pre-Merge UKF and the other Post-Merge UKF, differ in the way the feedback to the individual UKFs is applied. These novel methods have an advantage of less root mean square estimation error in position and velocity compared with the EKF and UKF and at the same time require much lesser number of computations than that of the PF, showing that these filters can serve as an optimal estimator. A testimony of the afore-mentioned advantages of the proposed novel methods is shown by carrying out Monte Carlo simulation in MATLAB R2009a for a typical war time scenario.

This paper develops a robust iterated extended Kalman filter (EKF) based on the generalized maximum likelihood approach (termed GM-IEKF) for estimating power system state dynamics when subjected to disturbances. The proposed GM-IEKF dynamic state estimator is able to track system transients in a faster and more reliable way than the conventional EKF and the unscented Kalman filter (UKF) thanks to its batch-mode regression form and its robustness to innovation and observation outliers, even in position of leverage. Innovation outliers may be caused by impulsive noise in the dynamic state model while observation outliers may be due to large biases, cyber attacks, or temporary loss of communication links of PMUs. Good robustness and high statistical efficiency under Gaussian noise are achieved via the minimization of the Huber convex cost function of the standardized residuals. The latter is weighted via a function of robust distances of the two-time sequence of the predicted state and innovation vectors and calculated by means of the projection statistics. The state estimation error covariance matrix is derived using the total influence function, resulting in a robust state prediction in the next time step. Simulation results carried out on the IEEE 39-bus test system demonstrate the good performance of the GM-IEKF under Gaussian and non-Gaussian process and observation noise.

Tennessee Eastman challenge process (TE) has been well known in process control literature as a challenge problem in estimation and control. Complexities in this nonlinear process and corresponding measurement models lead to non-Gaussian densities and estimators which can accommodate the non-Gaussianity can give better state estimates. In this work, we compare and demonstrate the utility of recently developed Sum of Gaussians based Unscented Gaussian Sum Filter (UGSF) for performing nonlinear state estimation of the process. UGSF uses sigma point concept to capture process nonlinearities coupled with a Sum of Gaussians approximation to achieve an accurate representation of non-Gaussian prior densities evolved from the process. The performance of UGSF is compared with the well known Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) approaches, for one of the operating modes of TE process. Results demonstrate the utility of using UGSF as a state estimator for the large-dimensional and highly nonlinear TE process.

This paper is motivated by the problem of more accurately estimating the hidden state in nonlinear dynamic system. We propose a uniform random sampling Kalman filter(URSKF) which can be regarded as a Sigma-point filter based on the statistical linearization method. Compared with other Sigma-point filters, the uniform random sampling method used to obtain the deterministic points can match the any-order moment of the prior distribution with the moderately increasing sampling points and automatically capture the more rich statistical properties of the system after nonlinear function mapping. Moreover, the URSKF is a derivative and square-rooting free operation which avoid computing Jacobian matrix and failing the filtering process for not always guaranteeing the covariance matrix to be positive definite, such as UKF. Besides, the computation complexity is linearly related to the system dimension avoiding the curse of high dimension in the Gauss-Hermite Quadratute Filter(GHQF). The performance of this filter is demonstrated by an aircraft tracking model. The simulation results show the URSKF performs higher accurately than the Unscented Kalman Filter(UKF), Central Difference Kalman Filter(CDKF) and Cubature Kalman Filter(CKF).

In this paper an offline approach for output-only Bayesian identification of stochastic nonlinear systems is presented. The approach is based on a re-parameterization of the joint posterior distribution of the parameters that define a postulated state-space stochastic model class. In the re-parameterization the state predictive distribution is included, marginalized, and estimated recursively in a state estimation step using an unscented Kalman filter, bypassing state augmentation as required by existing online methods. In applications expectations of functions of the parameters are of interest, which requires the evaluation of potentially high-dimensional integrals; Markov chain Monte Carlo is adopted to sample the posterior distribution and estimate the expectations. The proposed approach is suitable for nonlinear systems subjected to non-stationary inputs whose realization is unknown, and that are modeled as stochastic processes. Numerical verification and experimental validation examples illustrate the effectiveness and advantages of the approach, including: (i) an increased numerical stability with respect to augmented-state unscented Kalman filtering, avoiding divergence of the estimates when the forcing input is unmeasured; (ii) the ability to handle arbitrary prior and posterior distributions. The experimental validation of the approach is conducted using data from a large-scale structure tested on a shake table. It is shown that the approach is robust to inherent modeling errors in the description of the system and forcing input, providing accurate prediction of the dynamic response when the excitation history is unknown.

The unscented Kalman filter (UKF) is a widely used nonlinear Gaussian filter. It has the potential to deal with highly nonlinear dynamic systems, while displaying computational cost of the same order of magnitude as that of the extended Kalman filter (EKF). The quality of the estimates produced by the UKF is dependent on the tuning of both the parameters that govern the unscented transform (UT) and the two noise covariance matrices of the system model. In this paper, the tuning of the UKF is framed as an optimization problem. The tuning problem is solved by a new stochastic search algorithm and by a standard model-based optimizer. The filters tuned with the proposed algorithm and with the standard model-based optimizer are numerically tested against other nonlinear Gaussian filters, including two UKF tuned with state-of-the-art tuning strategies. One of these strategies relies on online tuning and the other on offline tuning.

This paper studies the consistency of the extended Kalman filter (EKF) for a kind of nonlinear systems. Based on the EKF algorithm, the authors propose the quasi-consistent EKF (QCEKF) as well as the tuning law for its parameters. The quasi-consistency of the proposed algorithm is proved. Finally, the feasibility of the algorithm is illustrated by the numerical simulation on an orbit determination example.

The state of charge (SOC) is a key indicator for the battery management system (BMS) of electric vehicles. A SOC joint estimation method based on the H infinity filter (HF) and unscented Kalman filter (UKF) algorithms is proposed in this paper, HF based parameters identification can trace the parameters online according to the working conditions while he UKF based state estimation method does not require the jacobian matrix derivation and the linearization for nonlinear model. The HF-UKF SOC joint estimation method has been experimentally validated at different temperatures. The results show that this method is robust to the inaccurate initial SOC value and the different working temperatures.

The iterated version of a family of non-linear Kalman filters, named the unscented transform (UT) based unscented Kalman filters (UKF), are revisited. Two existing frameworks of the iterated UKF are analysed and some shortcomings of them are pointed out. A new framework is proposed based on the statistical linear regression (SLR) perspective of the UT and the framework of the iterated extended Kalman filter (IEKF). The virtue of the proposed framework is twofold: first, the observation equation is linearised strictly following the SLR perspective implying that the regression error is also considered; second, it strictly follows the framework of the IEKF implying that in each iteration, the linearised equation is used to correct the a priori estimate rather than the latest estimate. A simple but illustrative benchmark example is simulated to check the feasibility of the proposed framework, and the results demonstrate the efficacy of the proposed framework.

The paper proves that the PDE dynamic model of the highway traffic is a differentially flat one and by applying spatial discretization it shows that the model’s transformation into an equivalent linear canonical state-space form is possible. For the latter representation of the traffic’s dynamics, state estimation is performed with the use of the Derivative-free nonlinear Kalman Filter. The proposed filter consists of the Kalman Filter recursion applied on the transformed state-space model of the highway traffic. Moreover, it makes use of an inverse transformation, based again on differential flatness theory which enables to obtain estimates of the state variables of the initial nonlinear PDE model. By avoiding approximate linearizations and the truncation of nonlinear terms from the PDE model of the traffic’s dynamics the proposed filtering methods outperforms, in terms of accuracy, other nonlinear estimators such as the Extended Kalman Filter. The article’s theoretical findings are confirmed through simulation experiments.

The aim of this work was to investigate the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and sequential importance resampling (SIR) filter for real-time estimation of a hyperbolic partial differential equation (PDE) system describing the gas flow transients in pipelines. The numerical method of lines was used for solving the system of PDEs. The spatial discretization was based on a five-point, fourth-order finite difference approximation and the time integrations were done using the fourth-order Runge-Kutta scheme (RK4). The resulting nonlinear state-space model was used by the Bayesian filtering algorithms for estimation of the system states. Numerical experiments were conducted to compare the accuracy, robustness and computation time. The results indicated that the difference in terms of accuracy between the EKF and UKF was small and considering the computation time of the UKF and numerically robustness of calculating the Jacobian matrix, the use of EKF might be a better choice when fast solutions are required. The UKF outperformed the EKF and SIR filter when steep variations in the mass flow rate boundary conditions occurred together with low model uncertainty.

Based on the concept of future measurement information, an unscented Rauch-Tung-Striebel smoother is developed to solve nonlinear inverse heat conduction problems. Numerical experiments are used to discuss the effect of the standard deviation of heat flux error, sampling time, sensor locations, the number of future time steps and the form of heat flux function on inversion results of the Rauch-Tung-Striebel smoother. The comparison with the unscented Kalman filter is conducted, and it shows that the Rauch-Tung-Striebel smoother greatly improves the time delay problem of the unscented Kalman filter, and reduces the sensitivity to measurement error. The inversion results of Kalman filter, extended Kalman filter, unscented Kalman filter and their corresponding smoothers for different nonlinear degrees are also compared, and it shows that for linear inverse problems, the Rauch-Tung-Striebel smoother is the optimal technology, for weak nonlinear inverse problems, the extended Rauch-Tung-Striebel smoother is a more appropriate choice, for the strong nonlinear inverse problems, the unscented Rauch-Tung-Striebel smoother is recommended.

Stochastic stability of a modified unscented Kalman filter is investigated for a class of nonlinear systems with stochastic nonlinearities and multiple fading measurements. Note that the stochastic nonlinearities are represented by statistical means which indicate multiplicative stochastic disturbances. The fading probability of fading measurements is individual for each sensor. Convergence on the modified unscented Kalman filter is established if there exists a lower bound for the mean of fading probability. Sufficient conditions are obtained to ensure stochastic stability of the modified unscented Kalman filter. The effectiveness of the filtering algorithm is verified by an illustrative numerical example.

New data filtering methods based on the Kalman filter concept are investigated for application to track geometry, where a track trolley is normally used to measure the track coordinates, cant and gauge, among others. Continuous as well as discrete measurements of the trolley are conceptually modeled, and the accuracy of the sampling scheme is also investigated. For this, a modified kinematic model of the track geometry involving transition and circular curves is proposed based on the tangential as well as normal acceleration, and both a nonlinear Extended Kalman Filter (EKF) and an augmented Unscented Kalman Filter (UKF) are applied to optimize the measured data. The efficiency and accuracy of the proposed models of EKF and UKF having a new kinematic equation is verified with ideal track geometry and with statistical methods including Root Mean Square (RMS). In addition, field measurement data are also considered to check the applicability of the models. Finally, future work on track geometry modeling based on a high-speed measurement vehicle is briefly outlined.

In order to eliminate the influence of outlier in measurement to X-ray pulsar based navigation system, a modified unscented Kalman filter-based filtering algorithm is proposed. First, the outlier d...