New approaches to Wiener filtering and Kalman filtering for descriptor systems (Q2726028)

The authors consider the following descriptor system: NEWLINE\[NEWLINEM x(t+1)=\Phi x(t)+\Gamma w(t),\quad y(t)=H x(t)+v(t),NEWLINE\]NEWLINE where \(x(t)\in {\mathbb R}^n\), \(y(t)\in {\mathbb R}^m\), \(M\) is singular and \(\det (zM-\Phi)\not\equiv 0, \forall z\in {\mathbb C}\), \(w(t)\) and \(v(t)\) are white noise with zero mean, and the system is completely observable. Using a decomposition of matrices and a time series analysis method, based on an autoregressive moving average (ARMA) innovation model and white noise theory, they present Wiener state estimators and steady-state Kalman estimators for the above-mentioned system. By the way, they handle the optimal filtering, smoothing, and prediction problems in a unified framework. According to their presented approach there is no need for solving Diophantine equations and Riccati equations. Two simulation examples are given to illustrate their approach and its effectiveness.