Modification of the Galerkin method for solving the problem of parametric synthesis of ACS

The objective of this paper is to develop and validate a modification of the generalized Galerkin method for solving the problem of parametric synthesis of nonlinear automatic control systems (ACS). Traditional methods focused on linear systems demonstrate insufficient efficiency when taking into account significant nonlinearities, which negatively affects the dynamic properties of controlled objects. The paper also focuses on the choice of a method for approximating nonlinear characteristics, where preference is given to a polynomial approximation, which provides a more accurate mathematical model compared to a piecewise linear approximation. Increasing the degree of the polynomial allows for a better match between the created model and the actual characteristics of the system. An algorithm for minimizing the objective function is proposed, taking into account the nonlinear properties and physical limitations of the systems. Recurrence relations for Galerkin integrals, which play a key role in determining the coefficients influencing the system dynamics, are obtained. The method is applicable to both continuous and pulse-based automatic control systems, including multivariate systems, significantly expanding its scope of practical application. An implementation algorithm has been developed, presented as a visual flowchart, including the steps of inputting initial data, converting normalized parameters into real values, calculating differential equation coefficients, checking system stability, and evaluating the solution's effectiveness by calculating the objective functional. Thus, a modification of the Galerkin method, combined with polynomial approximation of nonlinear characteristics, offers significant potential for improving the quality and reliability of control systems for various industrial facilities. The obtained results confirm the broad applicability of the proposed method to both continuous and pulse-based automatic control systems.