Kernel embedding-based state estimation for colored noise systems

In this paper, a novel methodology is proposed to solve the state estimation problem combining the extended Kalman filter (EKF) with a kernel recursive least squares (KRLS) algorithm (EKF-KRLS). The EKF algorithm estimates hidden states in the input space, while the KRLS algorithm estimates the measurement model. The algorithm works well without knowing the linear or nonlinear measurement model. We apply this algorithm to vehicle tracking, and compare the performances with traditional Kalman filter, EKF and KRLS algorithms. Results demonstrate that the performance of the EKF-KRLS algorithm outperforms these existing algorithms. Especially when nonlinear measurement functions are applied, the advantage of the EKF-KRLS algorithm is very obvious.

The Kalman filter (KF) is the optimal estimator that minimizes the mean square error when the state and measurement dynamics are linear in nature, provided the process and measurement noise processes are modeled as white Gaussian. However, in the real world, one encounters a large number of scenarios where either the process or measurement model (or both) are nonlinear. In such cases a class of suboptimal Kalman filter implementations called extended Kalman filters (EKF) are used. EKFs operate by linearizing the nonlinear model around the current reference trajectory and then designing the Kalman filter gain for the linearized model. Recently, an alternative approach has emerged for a certain class of problems where the error in the states is estimated using a Kalman filter, rather than the state itself. This error state KF (ErKF) approach, by deriving the error state dynamics, via the perturbation of the nonlinear plant, lends itself to optimal updates in the error states and optimal prediction and updates in the error state covariance. This is because the error state dynamics are linear, thereby satisfying a condition for optimal Kalman filtering. This paper offers a comparison between the EKF and ErKF via simulations and shows that the ErKF performance is robust to a variety of aircraft maneuvers performed. Furthermore, this paper shows that the ErKF, unlike the EKF, need not be repeatedly tuned with respect to the noise covariances in order to obtain acceptable estimation performance.

Traffic state estimation problems must deal with imperfect information. Imperfect information may occur as a result of misalignments between reality and the assumptions or simplifications made when developing the models as well as incomplete or corrupted data. One of the well-known methods of handling imperfect information is the Kalman filter algorithm and its variations, such as the extended Kalman filter and the unscented Kalman filter. While the standard Kalman filter and the extended Kalman filter have been widely applied to the state estimation of linear systems, the unscented Kalman filter has been reported the better choice in the multiclass traffic state estimation. In many applications of Kalman filters to traffic estimation problems, the model and measurement noise covariance matrices are normally estimated. When there is a mismatch between the true and the assumed noise distribution, however, the filter often suffers from performance degradation and even divergence in certain situations. To this end, this paper presents a more efficient and accurate algorithm embedded in the unscented Kalman filter to simultaneously estimate the traffic state and the model noise distribution statistics. The proposed method is facilitated through a simultaneous update of the model noise covariance matrix in the predicted covariance equations of the unscented Kalman filter algorithm. It is found through simulation that the proposed algorithm may improve the model performance over the standard algorithm.

The stochastic nature of noise signals affects the vehicle’s internal states and the outputs, resulting in the difficulty in estimation. The unknown or time-varying nature of noise signals if not taken into account for estimation, the results will diverge and be highly deteriorated. In this paper, a maximum likelihood principle (MLP) based adaptive robust extended kalman filter for estimating the states of the adopted non-linear vehicle model is proposed. An observability test is done for the purpose of estimation. A covariance matching (CM) based robust adaptive high forgetting factor based fault tolerant technique is also employed on the robust adaptive extended and unscented kalman filters for comparison purpose. The Robustness of the filter is analyzed by varying the parameter of the vehicle through a local sensitivity analysis. The results show that the MLP based approach to the extended kalman filter performs well in three simulations for sinusoidal steering, Double Lance Change, J-Turn, Fishhook, Slalom maneuver in comparison to robust adaptive unscented kalman filter. Friction coefficient of 0.8 (dry road) and 0.4 (wet road) is chosen for the simulation. The sideslip angle RMSE value for MLP based estimation is obtained as 2.62e-05, 4.545e-06 for Sine and DLC maneuver.

The paper presents a method of computational complexity reduction in Extended Kalman Filters dedicated for systems with non-linear measurement models. Extended Kalman filters are commonly used in radio-location and radio-navigation for estimating an object’s position and other parameters of motion, based on measurements, which are non-linearly related to the object’s position. This non-linearity forces designers to use non-linear filters, such as the Extended Kalman Filter mentioned, where linearization of the system’s model is performed in every run of the filter’s loop. The linearization, consisting of calculating Jacobian matrices for non-linear functions in the dynamics and/or observation models, significantly increases the number of operations in comparison to the linear Kalman filter. The method proposed in this paper consists of analyzing a variability of Jacobians and performing the model linearization only when expected changes of those Jacobians exceed a preset threshold. With a properly chosen threshold value, the proposed filter modification leads to a significant reduction of its computational burden and does not noticeably increase its estimation errors. The paper describes a practical simulation-based method of determining the threshold. The accuracy of the filter for various threshold values was tested for simplified models of radar systems.

Comparisons of six heating source localisation measurement models are conducted where temperature or ultrasonic time of flight readings provide the measurement update to the extended Kalman filter for estimating the location of a high heat flux spot source on a flat plate. For a particular measurement model, one of two processes are used: (1) directly using the measurements as the measurement vector in the extended Kalman filter or (2) indirectly obtaining the distance from the sensor to the heating source based on the measurement and then using the obtained distance as the measurement vector in the extended Kalman filter. For the direct models, the Jacobian required by the extended Kalman filter is obtained numerically using finite differences from the finite element forward conduction solution. For the indirect models, the derivatives of the distances with respect to the state variables are obtained in closed form. Heating source localisation results and convergence behaviour are compared for the six measurement models investigated. The ellipse from ultrasonic pulse one-way time of flight measurement model produces the best results when considering accuracy of converged solution, ability to converge to the correct solution given different initial guesses, and smoothness of convergence behaviour. Additionally, extended Kalman filter, extended information filter, and least squares inverse methods are compared for a parameter identification to quantify the heat flux and convection coefficient on the plate. All three inversion methods produce similar results which is significant as future work will consider only the extended Kalman filter.

The electrification of the transport sector increases the demand for a battery management system that allows the Li-ion battery to work in a sophisticated way to reach its maximum. The main focus of the battery management system is the estimation of the battery’s state of charge (SOC) which is an indicator to determine the driving range of an electric vehicle. However, battery modeling has a direct influence on SOC estimation. In this article, the equivalent circuit model (ECM) and extended Kalman filter (EKF) algorithm are combined for SOC estimation of the Li-ion battery. Since the performance of Kalman filter algorithms strictly depends on noise covariance matrices, incorrect values reduce the accuracy and make divergence in state estimation. Therefore, it is motivated to analyze the impact of process noise and measurement noise covariance matrix values on SOC estimation using Kalman filter algorithms. Ten different combinations of noise covariance matrices are employed to analyze the effect of these matrices on SOC estimation. As a result, the tuning of the filter is critical for improvement in state estimation regardless of the type of the battery model and filtering technique. The matrix values are chosen to take into account the convergence rate and estimation error. To validate the effectiveness of the chosen noise covariance matrices, the simulation results of the ECM based EKF estimation method is superimposed with the discharge curves available in the data sheet. The simulation results show minimal estimation error and a good convergence rate for tuned EKF.

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

This paper presents the efficient Unscented Kalman filter (UKF) based estimation and control of a 3-DOF Omni bundle robot. The angular trajectory tracking becomes more difficult under highly non-linear measurement noise and process noise. A rapidly converging Unscented Kalman Filter (UKF) is presented in the paper. The extended Kalman filter (EKF) provides less accurate results in the highly non-linear system and sometime may become unstable because of convergence related issues, whereas in UKF the results in the non-linear system are better than EKF. UKF is more accurate in state and noise estimation. The computed torque technique is taken as a model to apply UKF. After, estimation the noise is removed. The experiment is conducted on Omni bundle with process and measurement noise. The results prove the superiority of presented UKF over conventional UKF and EKF.

Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.

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.

Background and Objectives: Gene regulatory networks model the interactions among the genes and provide a decision rule describing activation and repression of each gene via various proteins. In order to be able to capture the complex gene interactions efficiently, it is imperative to develop algorithms that model the nonlinear interactions among the genes. This work considers the problem of inferring gene regulatory networks using time series data. A nonlinear model is assumed for the gene expression profiles, whereas the microarray data follows a linear Gaussian model. Methods: A particle filter based approach is proposed to estimate the gene expression profiles and the parameters are estimated online using the Kalman filter. In order to capture the inherent sparsity of gene networks, a least squares shrinkage selection operator (LASSO) based regression and model selection algorithm is proposed. Results: The performance of the aforementioned algorithm is rigorously evaluated for synthetic as well as real biological data sets arising from Drosophila melanogaster time series gene expression profiles. The results are contrasted with those reported in the literature. The performance of the proposed algorithm is compared with the extended Kalman filter (EKF) algorithm using mean square error (MSE) as the fidelity criterion. The proposed algorithm is observed to outperform the EKF in the scenarios considered. Conclusions: This work considered the problem of modeling and learning of gene regulatory networks using a nonlinear dynamical model. This represents a quite general modeling set-up. The gene network is modeled using a state space approach, and particle filtering is used for state estimation. The parameters regulating the interaction among genes are supplied by an online Kalman filter. Since the parameter vector is sparse, LASSO identifies the subset of these parameters pertaining to the relevant system coefficients. Extensive performance evaluations demonstrate that the proposed particle filter based approach outperforms EKF in terms of MSE.

Tracking a ballistic target in its reentry mode by considering the radar measurements is a highly complex problem in nonlinear filtering. Kalman Filter (KF) is used to estimate the position of target when the measurements are corrupted with noise. If the measurements are nonlinear (radar measurements) then Extended kalman filter (EKF) is used. For obtaining reliable estimate of the target state, filter has to be tuned before the operation which is offline. Tuning an EKF is the process of estimating the process noise covariance matrix (Q) and measurement noise covariance matrix (R). This paper presents a new method of tuning the EKF using different evolutionary algorithms.

This paper presents four vision-based tracking system architectures for marine surface objects using a fixed-wing unmanned aerial vehicle (UAV) with a thermal camera mounted in a pan/tilt gimbal. The tracking systems estimate the position and velocity of an object in the North-East (NE) plane, and differ in how the measurement models are defined. The first tracking system measures the position and velocity of the target with georeferencing and optical flow. The states are estimated in a Kalman filter. A Kalman filter is also utilized in the second architecture, but only the georeferenced position is used as a measurement. A bearing-only measurement model is the basis for the third tracking system, and because the measurement model is nonlinear, an extended Kalman filter is used for state estimation. The fourth tracking system extends the bearing-only tracking system to let navigation uncertainty in the UAV position affect the target estimates in a Schmidt-Kalman filter. All tracking architectures are evaluated on data gathered at a flight experiment near the Azores islands outside of Portugal. The results show that various marine vessels can be tracked quite accurately.

Today, induction motor (IM) is still the most popular electrical machine due to its robust and rare element-free structure, lower maintenance requirement, and cost-effective production. State estimation for this motor is the cornerstone for speed-sensorless control, fault-tolerant control, and fault diagnostics. Nonlinear Kalman filters, especially extended Kalman filters (EKFs), are the most preferred state and/or parameter estimation methods for IM. However, they require a stochastic system with complete process and measurement noise covariances for optimal estimations. These noise covariances, unknown or partially known in practice, vary under different operating conditions of the IM. To deal with this problem, various adaptive EKFs (AEKFs) have been proposed, which can compensate for the effect of varying noise covariances, but each approach has its own pitfalls. This article discusses the hybrid AEKF (HAEKF), which eliminates the problems of existing AEKFs. To demonstrate its effectiveness, the proposed HAEKF is compared qualitatively and quantitatively with existing AEKFs through simulation and experimental studies. Finally, improved estimation stability and performance are provided with the proposed HAEKF observer.