Analytical Synthesis of Quasi-Optimal Regulators for Linear Objects Based on Conditionally Adequate Low-Order Models. Part 1

Abstract

The complexity of solving the problem of synthesizing closed-loop control systems that are optimal in terms of rapid response multiply, rapidly increases with increasing order n of the control object — the well-known problem of the "curse of dimension" by R. Bellman. Therefore, for high-order linear objects (n l 4), exact, analytical control algorithms by the rapid response criterion are practically unknown, and for them approximate rapid response control laws are applied. The paper proposes an approach and, on its basis, develops analytical techniques for the synthesis of rapid response high-order systems, which use the idea of transforming the original performance problem to a similar control problem for objects of the first or second order, for which the optimal control algorithms are known. These algorithms underlie the proposed approach to the synthesis of rapid response control systems, which involves finding special functions describing the relationship between the phase coordinates of models of a low-order object (by analogy with the works of A. A. Kolesnikov, they are called aggregated or macrovariables) with the phase coordinates of the original high-order object, as well as the calculation of the parameters of the low-order models used, which provide, in a certain sense, their adequacy to the original object and, accordingly, the rapid response of the synthesized systems. The work distinguishes between the conditional and approximate adequacy of the models of the control object. In the case of conditional adequacy, the parameters of low-order models are found exactly using the eigenvalues and vectors of the original object, and with approximate adequacy, using the least squares method. In the first part of the work, two first-order models (conditionally and approximately adequate) are used, on the basis of which two methods of synthesis of rapid response controllers are developed. In the form of statements 1 and 2, the conditions for the applicability of these techniques are formulated. Accordingly, the main content of the first part of the work is devoted to the results of a comparative analysis of the properties of rapid response controllers obtained by using the proposed synthesis techniques. In the second part of the work, it is proposed to carry out a similar analysis of rapid response control systems synthesized using conditionally and approximately adequate models of a second-order object.