Synthesis of a Discrete Optimal Multidimensional Controller Based on Incomplete Data: Multidimensional Spectral Approach

The minimax formulation of the problem of linear stationary control based on incomplete data of multidimensional stationary in a broad sense random processes (vector useful signal) observed in an additive mixture with interference of the “white noise” type, when the spectral densities of disturbances in the measurement channel and in the measurement interference are completely unknown and belong to a certain set of non-negatively defined functions, is considered. Only the condition of linear regularity is imposed on the observed vector process. A guaranteeing estimate is considered, which means the best estimate of the parameters of a useful signal in the sense of a minimum standard error with the worst behavior of measurement errors and disturbances with spectral densities belonging to the set, with respect to which the optimal control is determined based on incomplete data. Regarding the spectral density of the useful signal, it is only known that it satisfies a given system of moment conditions and is concentrated on a given measurable subset of the frequency axis. It is shown that the factorization of the matrix spectral density makes it possible to obtain a solution to the problem of optimal minimax linear filtration and is necessary to solve the problem of linear optimal control based on incomplete data. The search for optimal control based on incomplete data from the emerging multidimensional antagonistic game is reduced to solving a specific system of relations. Matrix boundary value problem methods, Hilbert matrix transformations, and properties of matrix frequency characteristics are used in the solution. An illustrative example is presented.