ФУНКЦІЇ УОЛША В ЛІНІЙНО- КВАДРАТИЧНИХ ЗАДАЧАХ ОПТИМІЗАЦІЇ ЛІНІЙНИХ НЕСТАЦІОНАРНИХ СИСТЕМ

• Дата: 2025
• Автори: Stenin, A.A., Timoshin, Yu.A., Drozdovych, I.G.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Теми: лінійні нестаціонарні системи; лінійно-квадратичні оптимізаційні завдання; функції Уолша; фундаментальна матриця; замкнуте оптимальне керування
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/667
• Назва журналу: Problems of Control and Informatics

Currently, the solution of the problems of analytical design of the optimal controller (ADOC) for stationary dynamic objects is well studied and a number of works are devoted to them. At the same time, the synthesis of optimal control laws of non-stationary dynamic objects in general case is quite a complex task, which often can’t be solved in analytical form. This is primarily due to the difficulty of solving the nonstationary nonlinear vector-matrix Riccati equation. This article deals with linear-quadratic problems of synthesis of a closed optimal control law for one class of linear nonstationary systems. Determination of the optimal control law within the frame-work of the ADOC problem is based on the Pontryagin maximum principle. The fundamental matrix of the system of simplified canonical equations is used to establish the connection between the auxiliary vector and the state vector. In general case, it is not possible to obtain an analytical expression of the fundamental matrix for linear nonstationary systems. In this article it is proposed to find the fundamental matrix of the system of simplified canonical equations by means of approximate integration of the linear matrix differential equation of state, to which it satisfies, using the mathematical apparatus of Walsh functions. In this case, the elements of the matrix of the optimal control law are also determined in the form of Walsh series, the constant coefficients of which are found from the system of algebraic equations. Since the elements of the matrix of the optimal control law are piecewise constant functions, this greatly simplifies their practical implementation in comparison with the nonstationary matrices of optimal control obtained on the basis of the solution of the Riccati equation. The accuracy of the obtained approximate optimal solution is achieved by choosing the appropriate number of terms of the Walsh series expansion.