The behavioral approach to discrete-time deterministic Kalman filtering on the di-polynomial ring

In this paper, we address the discrete time deterministic Kalman filtering within a behavioral setting. In the continuous time case, Fagnani and Willems developed this issue by using the novel idea based on two-variable polynomial matrices and quadratic differential forms in [1] and [12]. We expand the Kalman filtering problem into the discrete time case by using two-variable di-polynomial matrices and quadratic difference forms studied in [3], [4] and [5]. Here we derive a sufficient condition for the latent variables of the filter to estimate the observed data and the state variable of the system in the sense that the sum of squared error signals between observed variables and estimated ones is minimized deterministically. By using this condition, we then clarify the structure of the optimal filter with respect to the notion of the state. And then we also provide the procedure for implementing the optimal filter as a real-time algorithm. Finally, we give an illustrative example in order to show the validity of our results.