A generalized autocovariance least-squares method for Kalman filter tuning (original) (raw)
Related papers
Power and Energy / 807: Intelligent Systems and Control / 808: Technology for Education and Learning, 2013
Designing a Kalman filter requires knowledge about the stochastic part of the system. Thus, disturbances affecting states and measurements should be known. However, in practical application these disturbances are usually unknown. In this contribution a modification of the autocovariance least-square method is presented. This method converts the measurement and process noise covariance estimation problem into a least squares functional, which can be solved with a Landweber iteration to regularize the illposed problem. Then, a tuned Kalman filter gain can be calculated. A simulative evaluation is introduced to prove the method regarding robustness against modeling error and variance of the estimates.
Two novel costs for determining the tuning parameters of the Kalman Filter
arXiv (Cornell University), 2011
The Kalman filter (KF) and the extended Kalman filter (EKF) are well established techniques for state estimation. However, the choice of the filter tuning parameters still poses a major challenge for the engineers [1]. In the present work, two new metrics have been proposed for determining the filter tuning parameters on the basis of the innovation covariance. This provides a metric based offline method usable for predicting the actual filter RMSE performances for a particular application and thus, for the selection of suitable combination(s) of the filter tuning parameters in order to ensure the design of a KF or an EKF having a balanced RMSE performance.
This report provides a brief historical evolution of the concepts in the Kalman filtering theory since ancient times to the present. A brief description of the filter equations its aesthetics, beauty, truth, fascinating perspectives and competence are described. For a Kalman filter design to provide optimal estimates tuning of its statistics namely initial state and covariance, unknown parameters, and state and measurement noise covariances is important. The earlier tuning approaches are reviewed. The present approach is a reference recursive recipe based on multiple filter passes through the data without any optimization to reach a 'statistical equilibrium' solution. It utilizes the a priori, a posteriori, and smoothed states, their corresponding predicted measurements and the actual measurements help to balance the measurement equation and similarly the state equation to help form a generalized likelihood cost function. The filter covariance at the end of each pass is heuristically scaled up by the number of data points is further trimmed to statistically match the exact estimates and Cramer Rao Bounds (CRBs) available with no process noise provided the initial covariance for subsequent passes. During simulation studies with process noise the matching of the input and estimated noise sequence over time and in real data the generalized cost functions helped to obtain confidence in the results. Simulation studies of a constant signal, a ramp, a spring, mass, damper system with a weak non linear spring constant, longitudinal and lateral motion of an airplane was followed by similar but more involved real airplane data was carried out in MATLAB R. In all cases the present approach was shown to provide internally consistent and best possible estimates and their CRBs. i ACKNOWLEDGEMENTS It is a pleasure to thank many people with whom the authors interacted over a period of time in the topic of Kalman filtering and its Applications. Decades earlier this topic was started as a course in the Department of Aerospace Engineering and a Workshop was conducted along with Prof. S. M. Deshpande who started it and then moved over full time to Computational Fluid Dynamics. Subsequently MRA taught the course and spent many years carrying out research and development in this area during his tenure at the IISc, Bangalore. The problem of filter tuning has always been intriguing for MRA since most people in the area tweaked rather than tuned most of the time with the result that there is no one procedure that could be used routinely in applying the Kalman filter in its innumerable applications. The PhD thesis of RMOG has been the only effort for a proper tuning of the filter parameters but this was not too well known. In the recent past for a couple of years the interaction among the present authors helped to reach the present method which we believe is quite good for such routine applications. The report has been written in such a way to be useful as a teaching material. Our grateful thanks are due to
Since the innovation of the ubiquitous Kalman filter more than five decades back it is well known that to obtain the best possible estimates the tuning of its statistics X 0 , P 0 , Θ, R and Q namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The earlier tweaking and other systematic approaches are reviewed but none has reached a simple and easily implementable approach for any application. The present reference recursive recipe based on multiple filter passes through the data leads to a converged 'statistical equilibrium' solution. It utilizes the pre, post, and smoothed state estimates and their corresponding measurements and the actual measurements as well as their covariances to balance the state and measurement equations and form generalized cost functions. The filter covariance at the end of each pass is heuristically scaled up by the number of data points and further trimmed to provide the P 0 for subsequent passes. A simultaneous and proper choice for Q and R based on the filter sample statistics and certain other covariances leads to a stable filter operation providing the results after few iterations. When only R is present in the data by minimizing the 'innovation' cost function J using the non filter based Newton Raphson optimization results served as an anchor for matching and tuning the filter statistics. When both R and Q are present in the data the consistency between the injected noise sequences and their statistics provided a simple route and confidence in the present approach. A typical simulation study of a spring, mass, damper system with a weak non linear spring constant shows the present approach out performs earlier techniques. The Part-2 of the paper further consolidates the present approach based on an analysis of real airplane flight test data.
2015
In the previous paper an adaptive ltering based on a reference recursive recipe was developed and tested on a simulated dynamics of a spring, mass, and damper with a weak nonlinear spring. In this paper the above recipe is applied to a more involved case of three sets of airplane data which have a larger number of state, measurements, and unknown parameters. Further the ight tests cannot always be conducted in an ideal situation of the process noise and the measurement noises being white and Gaussian as is generally assumed in the Kalman lter. The measurements are not available in general with respect to the center of gravity, possess scale and bias factors which will have to be modelled and estimated as well. The coupling between the longitudinal and lateral motion brings in added diculty but makes the problem more interesting. At times the noisy measurements from the longitudinal and lateral motion are input into the longitudinal states. This leads to the resulting equations becom...
A Review on Tuning of Extended Kalman Filter using Optimization Techniques for State Estimation
International journal of computer applications, 2016
State estimation is the common problem in every area of engineering. There are different filters used to overcome the problem of state estimation like Kalman filter, Particle filters etc. Kalman Filter is popular when the system is linear but when the system is highly non-linear then the different derivatives of Kalman Filter are used like Extended Kalman Filter (EKF), Unscented Kalman filter. But these estimation techniques require tuning of process and noise covariance matrices. The different optimization techniques are used to tune the filter parameters of EKF. In this paper, various optimization techniques have been studied for non-linear state estimation based on EKF.
Sādhanā
In part-1 of this paper an adaptive filtering based on a reference recursive recipe (RRR) was developed and tested on a simulated dynamics of a spring, mass and damper with a weak nonlinear spring. In this paper the above recipe is applied to a more involved case of three sets of airplane data that have a larger number of state, measurements and unknown parameters. The flight tests cannot always be conducted in an ideal situation of the process noise and the measurement noises being white Gaussian as is generally assumed in the Kalman filter. The measurements may not be available with respect to the center of gravity and possess scale and bias factors, which will have to be modelled and estimated as well. The coupling between the longitudinal and lateral motion brings in added difficulty but makes the problem more interesting. It turns out that even a parameter that strongly affects the airplane dynamics is estimated which vary widely among the approaches. The RRR has been shown to be better than the earlier approaches in estimating the unknowns. The generalized cost functions that are introduced in the present work help identify definitive results from deceptive results.
Tuning of Extended Kalman Filter for nonlinear State Estimation
IOSR Journal of Computer Engineering, 2016
Kalman Filter is the most popular method for state estimation when the system is linear. State estimation is the typical issue in every part of engineering and science. But, for non linear systems, different extensions of Kalman Filter are used. Extended Kalman Filter is famous to discard the non linearity which uses First order Taylor series expansion. But for these estimation techniques, the tuning of process noise covariance and measurement noise covariance matrices is required. There are different optimization techniques used to tune the parameters of Extended Kalman Filter. In this paper, Particle Swarm Optimization has been proposed to tune the EKF parameters and then the simulations are implemented for permanent magnet synchronous motor.
Optimal Covariance Minimization Algorithm for the Continuous Kalman Filter
AIAA/AAS Astrodynamics Specialist Conference, 2014
The classical Kalman filter algorithms obtain an optimal Kalman gain matrix by computing a stationary value for the covariance time derivative. This approach has proven to be extremely valuable for many engineering and scientific applications. The innovation of this work is that it develops a direct optimization approach for computing optimal Kalman gains. The resulting gain calculations rigorously minimize the a posteriori error covariance by computing a stationary value directly for the error covariance, as a function of correction gains for the filter. In addition, the resulting gain solutions directly minimize the measurement errors for the Filter. Algorithmic computational differentiation is used for generating the sensitivity partial derivatives required in the error covariance minimizing necessary conditions. A first-order correction strategy is presented for minimizing the elements of the error covariance matrix. A generalized covariance differential equation is developed that automatically generates the 2 nd through 4 th order moments for covariance, skewness, and Kurtosis, which are used to minimize the covariance matrix. The optimal Kalman gains are obtained numerically: no closed-form analytic solutions are available. Initial numerical experiments have been limited to Kalman gain sensitivity calculations. The basic methodology easily generalizes to handle state sensitivities for the plant and sensor. The proposed analysis approach is expected to be broadly useful for estimation and control problems, where model uncertainty is important for engineering level of fidelity applications.