Optimal Filtering and Its Information Version for Linear Discrete-Time Systems with Single Output Time-Delay

This paper investigates the optimal filtering and its information form for linear discrete-time systems with single time-delay in the observation signal. By introducing delay-free observation, the optimal filtering problem for time-delay systems can be converted into multiple steps prediction problem for delay-free systems. Then, the recursive form for optimal filter are given via innovation approach in Hilbert space. The calculation of the optimal filter involves solving one Riccati difference equation of the same dimensions as that of the original systems. The information filter can furthermore be designed by using the matrix inversion Lemma to optimal filtering formulas.