scholarly journals Kalman Filter Riccati Equation for the Prediction, Estimation, and Smoothing Error Covariance Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Nicholas Assimakis ◽  
Maria Adam
Keyword(s):  
Kalman Filter ◽  
Riccati Equation ◽  
Discrete Time ◽  
Prediction Error ◽  
Estimation Error ◽  
Riccati Equations ◽  
Covariance Matrices ◽  
Linear Estimation ◽  
Solution Algorithms ◽  
Error Covariance

The classical Riccati equation for the prediction error covariance arises in linear estimation and is derived by the discrete time Kalman filter equations. New Riccati equations for the estimation error covariance as well as for the smoothing error covariance are presented. These equations have the same structure as the classical Riccati equation. The three equations are computationally equivalent. It is pointed out that the new equations can be solved via the solution algorithms for the classical Riccati equation using other well-defined parameters instead of the original Kalman filter parameters.

scholarly journals Iterative and Algebraic Algorithms for the Computation of the Steady State Kalman Filter Gain

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Nicholas Assimakis ◽  
Maria Adam
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Prediction Error ◽  
Covariance Matrices ◽  
Asymptotically Stable ◽  
System Parameters ◽  
Estimation And Prediction ◽  
Error Covariance ◽  
Algebraic Algorithms ◽  
Time Invariant

The Kalman filter gain arises in linear estimation and is associated with linear systems. The gain is a matrix through which the estimation and the prediction of the state as well as the corresponding estimation and prediction error covariance matrices are computed. For time invariant and asymptotically stable systems, there exists a steady state value of the Kalman filter gain. The steady state Kalman filter gain is usually derived via the steady state prediction error covariance by first solving the corresponding Riccati equation. In this paper, we present iterative per-step and doubling algorithms as well as an algebraic algorithm for the steady state Kalman filter gain computation. These algorithms hold under conditions concerning the system parameters. The advantage of these algorithms is the autonomous computation of the steady state Kalman filter gain.

scholarly journals Model-Based Clustering with Measurement or Estimation Errors

Genes ◽  
2020 ◽  
Vol 11 (2) ◽  
pp. 185 ◽  
Author(s):  
Wanli Zhang ◽  
Yanming Di
Keyword(s):  
Finite Mixture Models ◽  
Estimation Error ◽  
Covariance Matrices ◽  
Finite Mixture ◽  
Estimation Errors ◽  
Data Set ◽  
Component Distribution ◽  
Model Based Clustering ◽  
Model Based ◽  
Error Covariance

Model-based clustering with finite mixture models has become a widely used clustering method. One of the recent implementations is MCLUST. When objects to be clustered are summary statistics, such as regression coefficient estimates, they are naturally associated with estimation errors, whose covariance matrices can often be calculated exactly or approximated using asymptotic theory. This article proposes an extension to Gaussian finite mixture modeling—called MCLUST-ME—that properly accounts for the estimation errors. More specifically, we assume that the distribution of each observation consists of an underlying true component distribution and an independent measurement error distribution. Under this assumption, each unique value of estimation error covariance corresponds to its own classification boundary, which consequently results in a different grouping from MCLUST. Through simulation and application to an RNA-Seq data set, we discovered that under certain circumstances, explicitly, modeling estimation errors, improves clustering performance or provides new insights into the data, compared with when errors are simply ignored, whereas the degree of improvement depends on factors such as the distribution of error covariance matrices.

Inflation Adjustment on Error Covariance Matrices for Ensemble Kalman Filter Assimilation

2013 ◽  
pp. 207-233
Author(s):  
Xiaogu Zheng ◽  
Guocan Wu ◽  
Xiao Liang ◽  
Shupeng Zhang
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Covariance Matrices ◽  
Error Covariance

scholarly journals On the Kalman Filter error covariance collapse into the unstable subspace

2011 ◽  
Vol 18 (2) ◽  
pp. 243-250 ◽  
Author(s):  
A. Trevisan ◽  
L. Palatella
Keyword(s):  
Kalman Filter ◽  
Lyapunov Exponents ◽  
Extended Kalman Filter ◽  
Covariance Matrices ◽  
Reduced Form ◽  
Asymptotic State ◽  
Numerical Verification ◽  
Error Covariance ◽  
Neutral Subspace ◽  
Unstable Subspace

Abstract. When the Extended Kalman Filter is applied to a chaotic system, the rank of the error covariance matrices, after a sufficiently large number of iterations, reduces to N+ + N0 where N+ and N0 are the number of positive and null Lyapunov exponents. This is due to the collapse into the unstable and neutral tangent subspace of the solution of the full Extended Kalman Filter. Therefore the solution is the same as the solution obtained by confining the assimilation to the space spanned by the Lyapunov vectors with non-negative Lyapunov exponents. Theoretical arguments and numerical verification are provided to show that the asymptotic state and covariance estimates of the full EKF and of its reduced form, with assimilation in the unstable and neutral subspace (EKF-AUS) are the same. The consequences of these findings on applications of Kalman type Filters to chaotic models are discussed.

Navigation of an Underwater Remotely Operated Vehicle Based on Extended Kalman Filter

2013 ◽  
Author(s):  
Blanca V. Martínez ◽  
Daniel A. Sierra ◽  
Rodolfo Villamizar
Keyword(s):  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Inertial Measurement Unit ◽  
Dynamic Models ◽  
Estimation Error ◽  
Covariance Matrices ◽  
Measurement Unit ◽  
Remotely Operated Vehicle ◽  
Filter Performance ◽  
Order Of Magnitude

An algorithm to estimate positions, orientations, linear velocities and angular rates of an Underwater Remotely Operated Vehicle (UROV), based on the Extended Kalman Filter (EKF), is presented. The complete UROV kinematic and dynamic models are combined to obtain the process equation, and measurements correspond to linear accelerations and angular rates provided by an Inertial Measurement Unit (IMU). The proposed algorithm is numerically validated and its results are compared with simulated UROV states. A discussion about the influence of the covariance matrices on the estimation error and overall filter performance is also included. As a conclusion, the proposed algorithm estimates properly the UROV linear velocities and angular rates from IMU measurements, and the noise in estimated states is reduced in about one order of magnitude.

scholarly journals UKF-Based Vehicle Pose Estimation under Randomly Occurring Deception Attacks

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Xinghua Liu ◽  
Dandan Bai ◽  
Yunling Lv ◽  
Rui Jiang ◽  
Shuzhi Sam Ge
Keyword(s):  
Kalman Filter ◽  
Pose Estimation ◽  
Upper Bound ◽  
Unscented Kalman Filter ◽  
Estimation Error ◽  
Accurate Estimation ◽  
Ground Vehicles ◽  
Ground Vehicle ◽  
Attack Model ◽  
Error Covariance

Considering various cyberattacks aiming at the Internet of Vehicles (IoV), secure pose estimation has become an essential problem for ground vehicles. This paper proposes a pose estimation approach for ground vehicles under randomly occurring deception attacks. By modeling attacks as signals added to measurements with a certain probability, the attack model has been presented and incorporated into the existing process and measurement equations of ground vehicle pose estimation based on multisensor fusion. An unscented Kalman filter-based secure pose estimator is then proposed to generate a stable estimate of the vehicle pose states; i.e., an upper bound for the estimation error covariance is guaranteed. Finally, the simulation and experiments are conducted on a simple but effective single-input-single-output dynamic system and the ground vehicle model to show the effectiveness of UKF-based secure pose estimation. Particularly, the proposed scheme outperforms the conventional Kalman filter, not only by resulting in more accurate estimation but also by providing a theoretically proved upper bound of error covariance matrices that could be used as an indication of the estimator’s status.

scholarly journals Comment on “Synergetic use of IASI and TROPOMI space borne sensors for generating a tropospheric methane profile product”

2021 ◽  
Author(s):  
Simone Ceccherini
Keyword(s):  
Kalman Filter ◽  
Data Fusion ◽  
Covariance Matrices ◽  
Complete Data ◽  
Fusion Method ◽  
Error Covariance ◽  
Noise Error

Abstract. A great interest is growing about methods that combine measurements from two or more instruments that observe the same species either in different spectral regions or with different geometries. Recently, a method based on the Kalman filter has been proposed to combine IASI and TROPOMI methane products. We show that this method is equivalent to the Complete Data Fusion method. Therefore, the choice between these two methods is driven only by the advantages of the different implementations. From the comparison of the two methods a generalization of the Complete Data Fusion formula, which is valid also in the case that the noise error covariance matrices of the fused products are singular, is derived.

On the efficient low cost procedure for estimation of high-dimensional prediction error covariance matrices

Automatica ◽  
2017 ◽  
Vol 83 ◽  
pp. 317-330 ◽  
Author(s):  
Hong Son Hoang ◽  
Remy Baraille
Keyword(s):  
Prediction Error ◽  
Low Cost ◽  
Covariance Matrices ◽  
High Dimensional ◽  
Error Covariance

Analisis Kestabilan Kalman Filter Sistem Singular dengan Pendekatan Deterministik

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Budi Rudianto
Keyword(s):  
Kalman Filter ◽  
State Space ◽  
Riccati Equation ◽  
Discrete Time ◽  
Simple Structure ◽  
Descriptor Systems ◽  
Deterministic Approach ◽  
Optimal Form

Makalah ini membahas Kalman filter dan Persamaan Aljabar Ricatti (PAR) untuk waktu diskrit. Lebih lanjut, sistem deskriptor varian waktu ditampilkan dalm bentuk formulasi umum. Pendekatan deterministik digunakan untuk menentukan bentuk optimum menjadi formulasi 9-block. Pernyataan 9-block selain menyatakan tahapan kondisi ruang, juga menampilkan struktur sederhana yang menarik dan simetris. Kemudian, kami akan menunjukkan bahwa Persamaan Aljabar Ricatti (PAR) memiliki semidefinit dan menstabilkan sistem.   In this paper will discuss the Kalman filter and Riccati equation for discrete-time. Furthermore, time-variant descriptor systems presented in a common formulation. Deterministic approach used to determine the optimal form into the formulation "9-block". The expression "9-block", besides stating stages pending state space, also presents a simple structure that is interesting and symmetrical. And then, we will show that the Aljabar Riccati Eqution has a stabilizing semi-definit.

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