Generalized LQR control and Kalman filtering with relations to computations of inner-outer and spectral factorizations

We investigate the generalized LQR control where the dimension of the control input is strictly greater than the dimension of the controlled output, and the weighting matrix on the control signal is singular. The dual problem is the generalized Kalman filtering where the dimension of the input noise process is strictly smaller than the dimension of the output measurement, and the covariance of the observation noise is singular. These two problems are intimately related to inner-outer factorizations for non-square stable transfer matrices with square inners of the smaller size. Such inner-outer factorizations are in turn related to spectral factorizations for power spectral density (PSD) matrices whose normal ranks are not full. We propose iterative algorithms and establish their convergence for inner-outer and spectral factorizations, which in turn solve the generalized LQR control and Kalman filtering.