A Chebyshev-like method for approximating matrix square root

2020 ◽  
Author(s):  
Sergio Amat ◽  
Sonia Busquier ◽  
José Antonio Ezquerro ◽  
Miguel Ángel Hernández-Verón ◽  
Ángel Alberto Magreñán
Keyword(s):  
Square Root ◽  
Matrix Square Root

scholarly journals Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Matthew Rhudy ◽  
Yu Gu ◽  
Jason Gross ◽  
Marcello R. Napolitano
Keyword(s):  
Sensor Fusion ◽  
Unscented Kalman Filter ◽  
Attitude Estimation ◽  
Square Root ◽  
Fusion Algorithm ◽  
Matrix Square Root ◽  
Global Positioning ◽  
The Matrix ◽  
Aerial Vehicle ◽  
Diagonalization Method

Using an Unscented Kalman Filter (UKF) as the nonlinear estimator within a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for attitude estimation, various methods of calculating the matrix square root were discussed and compared. Specifically, the diagonalization method, Schur method, Cholesky method, and five different iterative methods were compared. Additionally, a different method of handling the matrix square root requirement, the square-root UKF (SR-UKF), was evaluated. The different matrix square root calculations were compared based on computational requirements and the sensor fusion attitude estimation performance, which was evaluated using flight data from an Unmanned Aerial Vehicle (UAV). The roll and pitch angle estimates were compared with independently measured values from a high quality mechanical vertical gyroscope. This manuscript represents the first comprehensive analysis of the matrix square root calculations in the context of UKF. From this analysis, it was determined that the best overall matrix square root calculation for UKF applications in terms of performance and execution time is the Cholesky method.

A Non-linear and Noise-Tolerant ZNN Model and Its Application to Static and Time-Varying Matrix Square Root Finding

2018 ◽  
Vol 50 (2) ◽  
pp. 1687-1703
Author(s):  
Xiaoxiao Li ◽  
Jiguo Yu ◽  
Shuai Li ◽  
Zehui Shao ◽  
Lina Ni
Keyword(s):  
Time Varying ◽  
Square Root ◽  
Root Finding ◽  
Matrix Square Root ◽  
Non Linear ◽  
Noise Tolerant

scholarly journals A Second-Order Exact Ensemble Square Root Filter for Nonlinear Data Assimilation

2015 ◽  
Vol 143 (4) ◽  
pp. 1347-1367 ◽  
Author(s):  
Julian Tödter ◽  
Bodo Ahrens
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Second Order ◽  
Square Root ◽  
Computationally Efficient ◽  
Research Activities ◽  
Wide Range ◽  
Matrix Square Root ◽  
Non Gaussian ◽  
Ensemble Transform

Abstract The ensemble Kalman filter (EnKF) and its deterministic variants, mostly square root filters such as the ensemble transform Kalman filter (ETKF), represent a popular alternative to variational data assimilation schemes and are applied in a wide range of operational and research activities. Their forecast step employs an ensemble integration that fully respects the nonlinear nature of the analyzed system. In the analysis step, they implicitly assume the prior state and observation errors to be Gaussian. Consequently, in nonlinear systems, the analysis mean and covariance are biased, and these filters remain suboptimal. In contrast, the fully nonlinear, non-Gaussian particle filter (PF) only relies on Bayes’s theorem, which guarantees an exact asymptotic behavior, but because of the so-called curse of dimensionality it is exposed to weight collapse. Here, it is shown how to obtain a new analysis ensemble whose mean and covariance exactly match the Bayesian estimates. This is achieved by a deterministic matrix square root transformation of the forecast ensemble, and subsequently a suitable random rotation that significantly contributes to filter stability while preserving the required second-order statistics. The properties and performance of the proposed algorithm are further investigated via a set of experiments. They indicate that such a filter formulation can increase the analysis quality, even for relatively small ensemble sizes, compared to other ensemble filters in nonlinear, non-Gaussian scenarios. Localization enhances the potential applicability of this PF-inspired scheme in larger-dimensional systems. The proposed algorithm, which is fairly easy to implement and computationally efficient, is referred to as the nonlinear ensemble transform filter (NETF).

SSP Podded Propulsion Motor Control Based on SR-CDKF

2014 ◽  
Vol 986-987 ◽  
pp. 1142-1145
Author(s):  
Wen Long Yao
Keyword(s):  
Qr Decomposition ◽  
Estimation Accuracy ◽  
Square Root ◽  
Rotor Speed ◽  
Closed Loops ◽  
Speed Controller ◽  
Sensorless Vector Control ◽  
Matrix Square Root ◽  
The Matrix ◽  
Vector Control System

In this paper,the rotor speed and the position of the SSP propulsion motor are estimated for building sensorless vector control system with speed and current double closed loops based on square root center difference Kalman filter (SR-CDKF) algorithm. This method makes use of the QR decomposition linear algebra techniques and so on, and it updates the matrix square-root of the state covariance by the Cholesky factor updating. This method can not only get the more steady results but also improve the estimation accuracy of the SSP podded propulsion system. Simulation result shows that the improved CDKF algorithm is not only more accurate but also has higher rate of convergence compared with CDKF speed controller.

A note on computing the matrix square root

CALCOLO ◽  
2003 ◽  
Vol 40 (4) ◽  
pp. 273-283 ◽  
Author(s):  
B. Iannazzo
Keyword(s):  
Square Root ◽  
Matrix Square Root ◽  
The Matrix
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