A generalized eigenvalue problem for solving the discrete-time Riccati equation with singular dynamics and singular measurement noise

The discrete-time Riccati equation plays a central role in the solution of the discrete-time LQG control problem. An interesting aspect of the discrete-time Riccati equation, which is not shared by its continuous-time counterpart, is the fact that the equation may have a meaningful solution even if the measurement noise covariance is singular. These discrete-time singular measurement noise problems arise from sampled-data problems involving averaging A/D devices. These problems involve singular discrete-time plant dynamics as well. The purpose of this paper, therefore, is to extend the generalized eigenvalue approach to solve the discrete-time Riccati equation in the presence of both singular plant dynamics and singular measurement noise. The numerical method we develop allows arbitrary rank deficiency in the measurement noise covariance, including the extreme case of noiseless measurements.