Estimating Complex Signals With a Fuzzy-Enhanced Kalman Filter: A Note on “the Output Regulation and the Kalman Filter as the Signal Generator”

The authors recently introduced a Kalman filter-based approach for estimating noise-contaminated unmodeled signals. However, in that work, the dimension of the Kalman filter depends on the complexity of the signals being estimated, which are not known a priori. This typically requires a trial-and-error process, often resulting in an excessively large filter dimension that complicates the method’s practical application. To overcome this problem, this paper proposes a solution for estimating complex, unmodeled, and noise-contaminated signals while simultaneously obtaining a low-dimensional dynamical model whose output approximates the actual noise-free signal. This is achieved by incorporating a fuzzy error covariance matrix into the standard linear Kalman filter algorithm. The proposed modification prevents the Kalman filter estimate from drifting away from the true signal and provides an alternative to the traditional abrupt resetting of the error covariance matrix. Notably, this modification can be seen as a soft reset of certain initial conditions and, as demonstrated, does not compromise the Kalman filter’s stability. Furthermore, the result shows that incorporating simple fuzzy logic concepts may substantially enhance the performance of some established algorithms.