On an approach to filtering and control design for polynomial/delayed processes

This paper discusses the possibility to design the Kalman-Bucy optimal filter over observations with multiple time-varying delays. The procedure of designing the optimal filter over observations with single and multiple delays proceeding from the Ito-Volterra observation equations is proposed. The paper then presents the optimal nonlinear filter for quadratic state and linear observation equations confused with white Gaussian noises. The general scheme for obtaining the optimal filter in case of polynomial state and linear observation equations is announced. Using the duality principle, the optimal control problem is solved for a quadratic system with a linear control function and a quadratic cost criterion. The optimal controller is designed for an unobservable quadratic state with a linear control function, a quadratic cost criterion, and linear observations. The problem of uniting the obtained results and designing the optimal controller for polynomial states and linear observations with multiple time-varying delays is discussed in conclusion.