Gain-Free Square Root Information Filtering Using the Spectral Decomposition

A new square root state estimation algorithm is introduced, that operates in the information mode in both the time and the measurement update stages. The algorithm, called the V-Lambda filter, is based on the spectral decomposition of the covariance matrix into a V\V form, where V is the matrix whose columns are the eigenvectors of the covariance matrix, and A is the diagonal matrix of its eigenvalues. The algorithm updates a normalized state estimate along with the information matrix square root factors, thus doing away with the gain computation. Both stages of the filter constitute equation-free algorithms and thus ideally suit parallel processing implementations. Singular value decomposition is used as a sole computational tool in both the eigenvectors/eigenvalues and the normalized state estimate updates, rendering a complete estimation scheme with exceptional numerical stability and precision. The distinct square root nature of the new algorithm is demonstrated numerically via a typical example, which compares the performance of the V-Lambda filter to that of the corresponding conventional Kalman algorithm. Belonging to the class of square root estimation algorithms, the new filter has all the virtues of a true square root routine. However, the new formulation also provides its user with invaluable insight into the heart of the estimation process, which is a unique characteristic of the V-Lambda filters.