Generalized Stability Criteria for Special Linear Automatic Control Systems with a Time Delay

Abstract

Matters concerned with stability of special linear automatic control systems with a time delay are considered. The determining feature of such systems is that the control loop contains a link of pure (or transport) delay by time T (or for the path S) of the signal at its output with respect to the signal at its input. Apart from time delay links, these systems also contain other linear elements. Such systems include both longstudied systems for transporting various materials and relatively new selflearning repetitive control systems for accurately reproducing cyclically repeated motions or other signals with a period T (or S). The stability of special linear systems can be studied with fully resting on the Nyquist frequency criterion. According to the Nyquist frequency criterion, the stability of a closedloop system is judged by analyzing the frequency transfer function (FTF) loci of the openloop system that includes the controller and the controlled object with respect to the critical point (1, j0) on the complex plane. However, in the majority of cases, the presence of transcendental links results in that the FTF loci of the openloop system have the shape of a star with infinitely long rays, which makes it difficult to interpret the stability of systems with a time delay according to the Nyquist criterion. In such cases, it is more convenient to use a group of graphic criteria, the application of which makes it possible to establish sufficient stability conditions without examining the entire openloop system (the controller plus the object) by analyzing the relative position of the controlled object frequency response loci (obtained without taking into account the controller properties) and a certain stability region boundary, which is determined by the controller properties (obtained without taking into account the controlled object properties). To date, stability region boundaries have already been found for many types of specific structures of systems with a time delay. The problem is that an individually shaped stability region boundary has to be found for each specific type of controller. It is shown, based on a structural transposition of systems with a time delay, that their stability can be studied by using almost any stability boundaries that were previously proposed for some specific types of transcendental controllers. By generalizing the criteria it will be possible not only to estimate for the first time the stability of systems with a time delay in combination with a linear part of an arbitrary kind, but also to expand the variety of applied types of transcendental controllers.