Optimal nearly singular estimation of continuous linear stationary uniform rank systems

The problem of optimal estimation of linear continuous systems is considered for the case where all the noise signals are stationary white and the intensity of the measurement noise tends uniformly to zero. Explicit expressions for the estimation procedures and the corresponding minimum error covariance matrices of the estimate are obtained in terms of the measurement noise intensity and the zero properties of the system. These expressions are derived for the case where the first non-zero Markov parameter of the system is of full rank, for the infinite and the finite time filtering problems. They provide a simple geometric interpretation to the structure of the error covariance matrices and they distinctly show the difference between the estimation of minimum-and non-minimum-phase systems. The theory is developed both for square and non-square systems.