Kalman filtering for matrix estimation

A general discrete-time Kalman filter (KF) for state matrix estimation using matrix measurements is presented. The new algorithm evaluates the state matrix estimate and the estimation error covariance matrix in terms of the original system matrices. The proposed algorithm naturally fits systems which are most conveniently described by matrix process and measurement equations. Its formulation uses a compact notation for aiding both intuition and mathematical manipulation. It is a straightforward extension of the classical KF, and includes as special cases other matrix filters that were developed in the past. Beyond the analytical value of the matrix filter, it is shown through various examples arising in engineering problems that this filter can be computationally more efficient than its vectorized version.

[1]  Michael Athans,et al.  The Matrix Minimum Principle , 1967, Inf. Control..

[2]  Itzhack Y. Bar-Itzhack,et al.  Recursive Attitude Determination from Vector Observations: Direction Cosine Matrix Identification , 1984 .

[3]  E. Tse,et al.  A direct derivation of the optimal linear filter using the maximum principle , 1967, IEEE Transactions on Automatic Control.

[4]  Graham C. Goodwin,et al.  Adaptive filtering prediction and control , 1984 .

[5]  Paul S. Dwyer,et al.  Multivariate Maxima and Minima with Matrix Derivatives , 1969 .

[6]  Alex Berman,et al.  Optimal weighted orthogonalization of measured modes , 1979 .

[7]  H. Neudecker.  Some Theorems on Matrix Differentiation with Special Reference to Kronecker Matrix Products , 1969 .

[8]  Sabine Van Huffel,et al.  Total least squares problem - computational aspects and analysis , 1991, Frontiers in applied mathematics.

[9]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[10]  H. Weiss,et al.  Simple Structure for a High Performance Three-Dimensional Tracking Filter , 2004 .

[11]  J. Mendel.  Lessons in Estimation Theory for Signal Processing, Communications, and Control , 1995 .

[12]  Daniel Choukroun,et al.  Optimal-REQUEST Algorithm for Attitude Determination , 2001 .

[13]  Itzhack Bar-itzhack,et al.  On the Convergence of Iterative Orthogonalization Processes , 1976, IEEE Transactions on Aerospace and Electronic Systems.

[14]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[15]  James R. Wertz,et al.  Spacecraft attitude determination and control , 1978 .

[16]  Menahern Baruch,et al.  Optimal Weighted Orttiogonalization of Measured Modes , 1978 .

[17]  Gene H. Golub,et al.  An analysis of the total least squares problem , 1980, Milestones in Matrix Computation.

[18]  David H. Nissen.  A Note on the Variance of a Matrix , 1968 .

[19]  E. C. Macrae.  Matrix Derivatives with an Application to an Adaptive Linear Decision Problem , 1974 .

[20]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[21]  I. Bar-Itzhack.  REQUEST: A Recursive QUEST Algorithm for Sequential Attitude Determination , 1996 .