Influence Analysis of Unstable Zeroes and Poles on the Stability of the Feedback Systems

According to the theory of automatic control, there should be no behavior's difference between an object given by a set of transfer functions, which are respectively interconnected, and a real object, corresponding to such a theoretical structure with given transfer functions. Accordingly, a generalized analysis of the Otto Smith hypothesis regarding the stability indices in automatic control systems with unstable zeros and poles of second-order transfer functions is carried out. Due to the fact that the behavior of most technical objects can be described by a second-order transfer function, the main accent is placed on the second-order transfer function with a denominator with unstable zeros and poles. In the article, both the apparatus of transfer functions and the structural models of the appropriate level were used for the description, which made it possible to make their description evident. A generalized description of a second order automatic control system with negative feedback is made. For such a system, theoretical stability criteria have been formed with respect to its parameters on the basis of necessary and sufficient conditions of stability.

On the basis of the common description of the second-order transfer function, the study of automatic control systems with different variants of placement on the complex plane of unstable zeros and poles of the open system's transfer function was performed. The presentation of the material is accompanied by numerous examples, for which cases of transfer functions with both real poles and a pair of complex conjugated poles are considered. The case of both open system and feedback system is considered for each example given in the article. Both cases are illustrated in each example by bode plots and a step response.

The researches carried out in the article are illustrated by bode plots and step responses, which for each example are obtained using mathematical applications MATLAB (with the library Control   System   Toolbox)   and   Mathcad.   According   to   the   results   of   our   research,

O. Smith's conclusions about the difference in the behavior of real physical systems with unstable zeros and poles and theoretically obtained models with similar transfer functions are confirmed.