Calculation of the optimal filter in delayed perturbation systems

For an observed plant with delayed perturbations, an explicit formula for calculation of the linear quadratic filter in the Wiener–Kolmogorov problem was presented.The filter contains an integral over the delay interval. For replacement of the optimal filter by a suboptimal one designed disregarding the delay, an explicit expression for the increment of the performance functional was established. For small delays, the asymptotics was determined. For tracking a two-axle carriage whose model has a delay caused by the road imperfections acting first on the leading axle and then on the rear one, an example of filter calculation was presented.