Theory of Hidden Oscillations and Stability of Control Systems
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ILITY
Dedicated to the memory of Corresponding Member of the Russian Academy of Sciences Gennady Alekseevich Leonov
Theory of Hidden Oscillations and Stability of Control Systems N. V. Kuznetsov St. Petersburg State University, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected] Received May 13, 2019; revised August 30, 2019; accepted May 25, 2020
Abstract—The development of the theory of absolute stability, the theory of bifurcations, the theory of chaos, theory of robust control, and new computing technologies has made it possible to take a fresh look at a number of well-known theoretical and practical problems in the analysis of multidimensional control systems, which led to the emergence of the theory of hidden oscillations, which represents the genesis of the modern era of Andronov’s theory of oscillations. The theory of hidden oscillations is based on a new classification of oscillations as self-excited or hidden. While the self-excitation of oscillations can be effectively investigated analytically and numerically, revealing a hidden oscillation requires the development of special analytical and numerical methods and also it is necessary to determine the exact boundaries of global stability, to analyze and reduce the gap between the necessary and sufficient conditions for global stability, and distinguish classes of control systems for which these conditions coincide. This survey discusses well-known theoretical and engineering problems in which hidden oscillations (their absence or presence and location) play an important role. DOI: 10.1134/S1064230720050093
INTRODUCTION History of the theory of hidden oscillations. The mathematical modeling of the dynamics and determination of stability in technical systems is the most relevant direction in the scientific and technological development of any state that seeks to occupy a leading position in the modern world. The study of limiting dynamic regimes (attractors) and stability is necessary both in classical theoretical and in actual practical problems. One of the primary tasks of this kind is related to designing control systems: automatic regulators, in the XVIII–XIX centuries. Regulators had to ensure the transition of the dynamics of the control object to the operating mode and the stability of the operating mode relative to the initial deviations and external perturbations. A classic example is the Watt regulator, used to maintain a given constant speed of rotation of a turbine shaft. In Great Britain alone in 1868, about 75000 steam engines equipped with Watt regulators worked at industrial enterprises [1]. However, in the middle of the XIX century many of the steam engines produced were inoperative due to the phenomenon of self-oscillation, when the regulator could not cope with maintaining the alignment of the moments. The high accident rate of these machines posed new engineering requirements for their design and thereby stimulated the development of the mathematical theory
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