A differential algebraic parametrization of the Kalman/Luenberger observer gain

The present work revisits the Kalman/Luenberger observer design and proposes a parametrization of its observer gain through differential algebraic methods. In most practical cases observer gain is advantageously calculated through the so-called optimization methods consisting of solving a non-differential Riccati equation where the potential difficulties remain in the choice of weighting or uncertainty covariance matrices. In possible other situations it may be of interest to have at hand a straightforward parametrization of the observer gain. The unanswered theoretical question then is how to solve the under-determined equations specifying the observer gain in the multi-output case. Here it is proposed a solution which is purely based upon the differential algebraic geometry approach of observations problems which has been introduced for general nonlinear systems.