Operational-optimal finite-dimensional dynamic controller of the stochastic differential plant’s state according to its output. II. Linear-Gaussian plant with inaccurate state measurements and a quadratic-biquadratic criterion

This article presents a general algorithm for the synthesis of an optimal, on average, fast dynamic regulator, performed sequentially in time and obtained in the first part, with a selectable order for the case of inaccurate measurements of the output of a nonlinear stochastic controlled object. Its application to a particular control problem for a linear-Gaussian object with a variable quality criterion, which is quadratic in the control and regulator state but quadratic-biquadratic in the object state, is demonstrated. It is shown that the optimal nonlinear structural functions of the controller state equation and its output formula in this case are expressed in terms of the first three initial moments of the conditional probability density obtained by solving the Cauchy problem for a nonlinear partial differential equation obtained from the Fokker–Planck–Kolmogorov equation. To find the approximate analytical form of these functions, the Gaussian approximation method is used, which reduces the problem to obtaining the coefficients of some nonlinearities, which depend only on time, by the sequential Monte Carlo method. It is shown that the biquadraticity of the criterion leads to a useful polynomial form up to the third degree for the structural functions of the Gaussian regulator, while for the quadratic criterion, a linear regulator satisfying Wonham’s separation theorem is obtained, and its order is equal to the object order.