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Theory of Hidden Oscillations and Stability of Control Systems

  • STABILITY
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Journal of Computer and Systems Sciences International Aims and scope

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.

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Notes

  1. A.A. Witt was a coauthor of the first edition of the book [8], published in 1937, but his name was removed from the first edition and restored only in the second edition of the book, published in 1959. In 1941, a monograph by I. Rockard, one of the creators of the atomic bomb in France, was published with a similar name and similar ideas in French [9] without reference to the works of Andronov.

  2. The concept of self-excitation of oscillations in Andronov’s works was also used to describe the bifurcation process of the transition of the system’s state to the oscillation mode with changing parameters [8].

  3. An attractor of a dynamical system is a bounded closed invariant set in phase space that is locally attractive (i.e., having an open neighborhood—a basin of attraction, all trajectories with the initial data from which tend to the attractor over time).

  4. Several paragraphs in the monograph [7] are devoted to the corresponding analytical analysis.

  5. G. Barkhausen used the similar German term Selbsterregte Schwingungen in his papers [44].

  6. https://scholar.google.com/scholar?q=hidden+attractor

  7. This doctoral thesis was defended at St. Petersburg State University in 2016 (reviewers: I.M. Burkin, N.G. Kuznetsov, G.A. Leonov, E.A. Mikrin, V.G. Peshekhonov, R.M. Yusupov, V.I. Nekorkin, and A.M. Sergeev (Leading Organization–Institute of Applied Physics of the Russian Academy of Sciences)).

  8. RAS, news 11.12.2016 (Russian Highly Cited Researchers Award, 2016): http://www.ras.ru/news/shownews.aspx?id=036a64c2-32f2-4624-bc32-8f0e4d138e7d. See the citation of the review on latent oscillations in the translated version of Izvestia RAN. Control theory and systems: http://citations.springer.com/search?query=Computer+and+Systems+Sciences+International.

  9. The corresponding rigorous mathematical statements were first formulated later for the general case of continuous systems by E.A. Barbashin and N.N. Krasovsky [49].

  10. Note that we can construct counterexamples to the discrete analogue of the Kalman conjecture for 2-dimensional systems [53]. Here, the difference in the phase space dimension needed for constructing counterexamples for continuous and discrete cases coincides with the difference in the dimension of the spaces of dynamical systems in which chaos occurs: 3 and 1, respectively.

  11. Here the solution of the discontinuous system is understood in the terms of Filippov [56]; the function \(\tanh(100\sigma )\) is used as a smooth approximation sign(σ).

  12. In the two-dimensional case, the departure of the trajectories is possible only to infinity, and in the three-dimensional case, the existence of limit periodic trajectories is possible [59].

  13. Regarding counterexamples with hidden attractors [41, 67] in 2011, R.E. Kalman wrote to the author of the article: “I was far too young and lacking technical mathematical knowledge to go more deeply into the matter.”

  14. In physical experiments, the system’s state leaves the unstable stationary mode due to external perturbations (an example is the impossibility of observing the upper position of the physical pendulum without additional stabilization). When analyzing the corresponding mathematical dynamic models, it is necessary to take into account that the unstable equilibrium states themselves do not fall into the basin of attraction of self-excited attractors. Therefore, in numerical modeling, the system state can remain in an unstable equilibrium state, and to study the dynamics in its vicinity, one has to choose the initial data that are different from the equilibrium state itself.

  15. In 2009, the plenary report [92] needed an example of applying the harmonic balance method to the Chua electronic circuit. For the parameters that I randomly selected, the attractor found in the Chua chain turned out to be weakly connected to the equilibrium states. A small additional control made it possible to disconnect the attractor found from the equilibrium states. In 2010, this example was finalized and presented at the IFAC conference “Periodic Control Systems” [40], and also published in the journal “Proceedings of the Academy of Sciences” [93]. Then, using the hidden attractor in the modified circuit, using the parameter continuation method, we managed to get rid of the additional control and get the hidden attractor in the classical Chua chain in 2011.

  16. This phase portrait with three Chua hidden attractors was selected for the cover of the International Journal of Bifurcation and Chaos in Applied Sciences and Engineering: https://www.worldscientific.com/na101/home/literatum/publisher/wspc/journals/content/ijbc/2017/ijbc.27.issue-12/ijbc.27.issue-12/20171218/ijbc.27.issue-12.cover.jpg. The initial data for visualizing hidden attractors in system (3.1): \(z = (9.2942,5.5013, - 31.4277)\), \(z = \pm (1.5179,0.2875, - 1.7414)\).

  17. Academician of the Academy of Sciences of the USSR (since 1946), rector of Moscow State University (1951–1973).

  18. Designing PLL systems for the analysis of stability and oscillations, various software simulators of electronic systems, Simulation Program with Integrated Circuit Emphasis (SPICE), are used by engineers, giving the illusion of virtual reality—observing real physical processes.

  19. In 1986, G.A. Leonov (as part of a team of researchers led by V.V. Shahgildyan) was awarded the USSR State Prize for these works.

  20. In 2008–2009 the coleaders were G.A. Leonov and V.A. Yakybovich (NSh-2387.2008.1), in 2014–2017 the coleader was G.A. Leonov (NSh-3384.2014.1, and NSh-8580.2016.1), and since 2018, the team of the Leading Scientific School of the Russian Federation has been headed by N.V. Kuznetsov (NSh-2858.2018.1 and NSh-2624.2020.1).

  21. Materials of the report “Hidden oscillations and stability of control systems. Theory and Applications” at the meeting of the Bureau of the DEEMCP on March 3, 2020: http://apcyb.spbu.ru/wp-content/uploads/2020-RAN-Bureau-DEEMCP-KuznetsovNV.pdf

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Funding

This work was supported by the Russian Science Foundation (project 19-41-02002, 2019-2021).

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  1. St. Petersburg State University, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    N. V. Kuznetsov

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Correspondence to N. V. Kuznetsov.

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Dedicated to the memory of Corresponding Member of the Russian Academy of Sciences Gennady Alekseevich Leonov

Translated by E. Seifina

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Kuznetsov, N.V. Theory of Hidden Oscillations and Stability of Control Systems. J. Comput. Syst. Sci. Int. 59, 647–668 (2020). https://doi.org/10.1134/S1064230720050093

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