Connection between the three-block generalized Riccati equation and the standard Riccati equation

Nikoukhah et al. (1992) developed a maximum likelihood descriptor Kalman filter the properties of which are governed by a three-block generalized Riccati difference equation (GRDE). The convergence of this GRDE has been studied and the generalization of the eigenvector approach to solving the associated generalized algebraic Riccati equation has been carried out. In this paper, the authors connect the GRDE to the standard Riccati equation to point out Nikoukhah´s results from well-known results obtained in standard Kalman filtering. Moreover, while the previous study provides only sufficient conditions on the convergence and stability of the three-block descriptor Kalman filter, the relationship established herein allows the authors to derive necessary and sufficient conditions. In addition, several descriptor Kalman filtering problems can be treated as in the standard case