Equivalence of Models of Minimally-Phase Linear Automatic Control Systems with Pid-Regulators in a Non-Minium Phase

Abstract

For closed automatic control systems of linear minimum phase dynamic objects with PID controllers, the processes in which are described by mathematical models in the form of ordinary linear differential equations of high order, the article proposes a method for synthesizing mathematical models in the form of differential equations of not higher than the second order in the class of nonminimum phase ones, i.e. in the form of differential models of not higher than the second order with an argument which delays for a period of passing an input signal of the system to its output.

The article also presents a method for identifying equivalent models based on the transferring of mathematical models defined on the time domain to the frequency domain.

To derive the calculated ratios of the proposed method of identifying equivalent models, there were used the frequency characteristics.

As a criterion for optimizing parameters of the equivalent models, the criterion of least squares was used.

As the main constraint in solving the problem of finding the optimal values of parameters of the equivalent models, there was chosen an equality of the critical frequencies of the equivalent system and the initial one, because it is the value of the critical frequency that determines the ability of a linear dynamical system to retain or lose stability when it is closed by a unity negative feedback, and therefore, when equivalenting, it is necessary for the equivalent model to set the same value of the critical frequency that the real dynamic system has, for which equivalenting is used.