Solution in the z-domain to the discrete-time stationary Kalman filtering problem for systems with perfect measurements

The discrete-time stationary Kalman filtering problem is solved in the
z
-domain for general right invertible linear systems whose measurements are all noise free. A simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases. It provides an optimal solution also for systems with uniform and nonuniform time delays. The minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle, finite and infinite, and the excess of the number of inputs over the number of measurements are investigated.