Transfer functions of a time-varying control system
Volt Avdieiev
Oles Honchar Dnipro National University
The mathematical model of linear time-varying systems (LTV) is the differential equations with coefficients that change over time. The question of their analysis and synthesis is an integral part of the control theory, the development of which is caused by the need to solve a number of technical problems, in particular, the design of aircraft motion control systems. For the study of LTV, in particular, with the aim of synthesizing the regulation law that ensures the specified indicators, various variants of the mathematical apparatus have been used, but the methods of applied value in order to determine the LTV dynamic characteristics have not been given due attention. The goal of the paper is to develop methodological support for constructing an algorithm for determining the equivalent stationary approximation, that is, of the transfer function which is equivalent to the LTV at the selected time interval. The task is to show the possibility of obtaining a second-order transfer function, which is equivalent to the LTV on a certain trajectory section, using the example of the system for controlling the rotational motion of a rocket in one plane. The concept of a transfer function is based on the integral Laplace transform, which maps the time function to complex argument function. This makes it possible to transform differential equations into algebraic ones, as a result to use the developed apparatus of linear algebra to solve problems of analysis and synthesis. In the work, the time-varying component of the model parameters is presented in the form of the sum of a finite number of exponential functions, which significantly simplifies the Laplace transformation algorithm of LTV differential equations. This makes it possible for the selected time interval to construct the stationary equivalent of the LTV in the form of a transfer function, which can be used to estimate the stability margin, the type and duration of the transient process of disturbance compensation, and the frequency response.
