Optimal and robust integral sliding mode filter design for systems with continuous and delayed measurements

In this paper, the optimal filtering problem for a linear system over observations with a fixed delay is treated, proceeding from the general expression for the stochastic Ito differential of the optimal estimate and its variance. As a result, the optimal filtering equations similar to the traditional Kalman-Bucy filter are obtained in the form dual to the Smith predictor, commonly used control structure for model-based time delay compensation. The paper then presents a robustification algorithm for the obtained optimal filter based on sliding mode compensation of disturbances. As a result, the sliding mode control of observations leading to suppression of the disturbances in a finite time is designed. This control algorithm also guarantees finite-time convergence of the estimate based on the corrupted observations to the optimal estimate satisfying the obtained optimal filtering equations over delayed observations.