Hi Zym,
The filter function in MATLAB's System Identification Toolbox uses a Kalman filter to estimate the state of a dynamic system based on noisy measurements. The Kalman filter involves computing the inverse of a covariance matrix, which can become singular if the matrix is ill-conditioned or poorly conditioned. To avoid this issue, the filter function regularizes the covariance matrix by adding a small amount of noise to the diagonal elements. This is known as "covariance inflation" or "covariance regularization". The amount of noise added is controlled by the Tolerance parameter, which specifies the maximum allowable ratio of the largest to smallest eigenvalue of the covariance matrix. If this ratio exceeds the tolerance, the diagonal elements of the covariance matrix are increased until the ratio falls below the tolerance.
The filter function does not use Cholesky inversion for the correction. Instead, it uses a modified version of the Kalman filter that incorporates the covariance inflation. The exact details of the algorithm are not disclosed in the documentation, as it is an in-built function.
Hope this helps!
