ESTIMATION OF COVARIANCES FOR KALMAN FILTER TUNING USING AUTOCOVARIANCE METHOD WITH LANDWEBER ITERATION

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.

[1]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[2]  Claudio Estatico,et al.  Improvement of Space-Invariant Image Deblurring by Preconditioned Landweber Iterations , 2008, SIAM J. Sci. Comput..

[3]  D. G. Watts,et al.  Spectral analysis and its applications , 1968 .

[4]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[5]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[6]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[7]  Bertolt Eicke.  Iteration methods for convexly constrained ill-posed problems in hilbert space , 1992 .

[8]  M. Bertero,et al.  Projected Landweber method and preconditioning , 1997 .

[9]  Thomas S. Huang,et al.  Unified Hilbert space approach to iterative least-squares linear signal restoration , 1983 .

[10]  James B. Rawlings,et al.  Estimation of the disturbance structure from data using semidefinite programming and optimal weighting , 2009, Autom..

[11]  James B. Rawlings,et al.  The autocovariance least-squares method for estimating covariances: application to model-based control of chemical reactors , 2006, IEEE Transactions on Control Systems Technology.

[12]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[13]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[14]  Brian J. Odelson.  Estimating Disturbance Covariances From Data For Improved Control Performance , 2003 .

[15]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[16]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice Using MATLAB , 2001 .

[17]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[18]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice , 1993 .

[19]  Bodo Heimann,et al.  A multi-axial electromagnetically actuated punch for cutting micro-components , 2012, 2012 American Control Conference (ACC).

[20]  Greg Welch,et al.  Welch & Bishop , An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960 , 1994 .