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 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 Global Systems for Mobile Position Tracking Using Kalman and Lainiotis Filters

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Nicholas Assimakis ◽  
Maria Adam
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Impulse Response ◽  
Finite Impulse Response ◽  
Position Tracking ◽  
Computational Burden ◽  
Time Invariant ◽  
Global Systems

We present two time invariant models for Global Systems for Mobile (GSM) position tracking, which describe the movement inx-axis andy-axis simultaneously or separately. We present the time invariant filters as well as the steady state filters: the classical Kalman filter and Lainiotis Filter and the Join Kalman Lainiotis Filter, which consists of the parallel usage of the two classical filters. Various implementations are proposed and compared with respect to their behavior and to their computational burden: all time invariant and steady state filters have the same behavior using both proposed models but have different computational burden. Finally, we propose a Finite Impulse Response (FIR) implementation of the Steady State Kalman, and Lainiotis filters, which does not require previous estimations but requires a well-defined set of previous measurements.

Estimation of Steady-State Optimal Filter Gain From Nonoptimal Kalman Filter Residuals

1994 ◽  
Vol 116 (3) ◽  
pp. 550-553 ◽  
Author(s):  
Chung-Wen Chen ◽  
Jen-Kuang Huang
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Finite Number ◽  
Discrete Time ◽  
Stochastic System ◽  
Moving Average ◽  
Ar Model ◽  
Numerical Example ◽  
Measurement Noises ◽  
Time Invariant

This paper proposes a new algorithm to estimate the optimal steady-state Kalman filter gain of a linear, discrete-time, time-invariant stochastic system from nonoptimal Kalman filter residuals. The system matrices are known, but the covariances of the white process and measurement noises are unknown. The algorithm first derives a moving average (MA) model which relates the optimal and nonoptimal residuals. The MA model is then approximated by inverting a long autoregressive (AR) model. From the MA parameters the Kalman filter gain is calculated. The estimated gain in general is suboptimal due to the approximations involved in the method and a finite number of data. However, the numerical example shows that the estimated gain could be near optimal.

scholarly journals Two-Sample Kalman Filter for Steady-State Data Assimilation

2008 ◽  
Vol 136 (11) ◽  
pp. 4503-4516 ◽  
Author(s):  
Julius H. Sumihar ◽  
Martin Verlaan ◽  
Arnold W. Heemink
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Data Assimilation ◽  
Simple Wave ◽  
Wave Model ◽  
State Data ◽  
Error Covariance ◽  
Observation Network ◽  
Kalman Gain ◽  
Potential Applicability

Abstract In this paper, a new iterative algorithm for computing a steady-state Kalman gain is proposed. This algorithm utilizes two model forecasts with statistically independent random perturbations to determine the error covariance used to define a Kalman gain matrix for steady-state data assimilation. It is based on the assumption that the error process is weakly stationary and ergodic. The algorithm consists of an iterative procedure for improving the covariance estimate, which requires a fixed observation network. Two twin experiments using a simple wave model and an operational storm surge prediction model are performed to demonstrate the performance of the proposed algorithm. The experiments show that the results obtained by using the proposed algorithm converge to the ones produced by the classic Kalman filter algorithm. An additional experiment using the three-variable Lorenz model is also performed to demonstrate its potential applicability in unstable dynamical systems.

The Design of a Robust Weighted Measurement Fusion Steady-State Kalman Filter

2014 ◽  
Vol 701-702 ◽  
pp. 624-629
Author(s):  
Wen Qiang Liu ◽  
Xue Mei Wang ◽  
Zi Li Deng
Keyword(s):  
Kalman Filter ◽  
Steady State ◽  
Lyapunov Equation ◽  
Model Parameters ◽  
Invariant System ◽  
Parameter Uncertainties ◽  
Worst Case ◽  
Uncertain Model ◽  
Time Invariant ◽  
Time Invariant System

For the linear discrete-time multisensor time-invariant system with uncertain model parameters and measurement noise variances, by introducing fictitious noise to compensate the parameter uncertainties, using the minimax robust estimation principle, based on the worst-case conservative multisensor system with conservative upper bounds of measurement and fictitious noises variances, a robust weighted measurement fusion steady-state Kalman filter is presented. By the Lyapunov equation approach, it is proved that when the region of the parameter uncertainties is sufficient small, the corresponding actual fused filtering error variances are guaranteed to have a less-conservative upper bound. Simulation results show the effectiveness and correctness of the proposed results.

Steady-State Optimal State and Input Observer for Discrete Stochastic Systems

1989 ◽  
Vol 111 (2) ◽  
pp. 121-127 ◽  
Author(s):  
Y. Park ◽  
J. L. Stein
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Stochastic Systems ◽  
Linear Time ◽  
The State ◽  
Optimal State ◽  
Unknown Inputs ◽  
Linear Time Invariant ◽  
Error Covariance ◽  
Time Invariant

Model-based machine diagnostics techniques require the modeled states and machine inputs to be measured. Because measurement of all the states and inputs is not always possible or practical, a simultaneous state and input observer is required. Previous work has developed this type of acausal observer and shown it is susceptible to noise. This paper develops a steady-state optimal observer that minimizes the trace of the steady-state error covariance of the state and input estimates for discrete, linear, time-invariant, stochastic systems with unknown inputs. In addition, a method to distinguish the best measurement set among the available measurement sets is developed. Results from numerical simulations show that the optimal observer can greatly improve estimation results in some cases.

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.

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.

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