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Journal of Computer and Systems Sciences International

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01.09.2020 | STABILITY

Theory of Hidden Oscillations and Stability of Control Systems

verfasst von: N. V. Kuznetsov

Erschienen in: Journal of Computer and Systems Sciences International | Ausgabe 5/2020

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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.

Nächster Artikel Development and Research of Selectively Invariant Electromechanical Systems with the Adaptation of Regulators to Velocity Level Changes

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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.

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].

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).

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

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

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

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)).

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.

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].

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.

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).

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].

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.”

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.

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.

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): \(z = (9.2942,5.5013, - 31.4277)\), \(z = \pm (1.5179,0.2875, - 1.7414)\).

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

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.

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.

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).

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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M. A. Kiseleva, E. V. Kudryashova, N. V. Kuznetsov, O. A. Kuznetsova, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Hidden and self-excited attractors in Chua circuit: Synchronization and SPICE simulation,” Int. J. Parall., Emerg. Distrib. Syst. 33, 513–523 (2018).CrossRef

100.

A. Sommerfeld, “Beitrage zum dynamischen Ausbau der Festigkeitslehre,” Zeitschr. Vereins deutsch. Ingen. 46, 391–394 (1902).

101.

I. I. Blekhman, “Self-synchronization of vibrators of some vibrating machines,” Inzh. Sb. 16, 49–72 (1953).

102.

M. A. Kiseleva, N. V. Kuznetsov, and G. A. Leonov, “Hidden attractors in electromechanical systems with and without equilibria,” IFAC-PapersOnLine 49 (14), 51–55 (2016).CrossRef

103.

M. Kiseleva, N. Kondratyeva, N. Kuznetsov, and G. Leonov, “Hidden oscillations in electromechanical systems,” in Dynamics and Control of Advanced Structures and Machines, Ed. by H. Irschik, A. Belyaev, and M. Krommer (Springer, Cham, 2018), pp. 119–124.

104.

I. I. Blekhman, D. A. Indeitsev, and A. L. Fradkov, “Slow motions in systems with inertially excited vibrations,” IFAC Proc. Vols. 40 (14), 126–131 (2007).

105.

A. Fradkov, O. Tomchina, and D. Tomchin, “Controlled passage through resonance in mechanical systems,” J. Sound Vibrat. 330, 1065–1073 (2011).CrossRef

106.

N. Mihajlovic, A. A. van Veggel, N. van de Wouw, and H. Nijmeijer, “Analysis of friction-induced limit cycling in an experimental drill-string system,” J. Dyn. Syst. Meas. Control 126, 709–720 (2004).CrossRef

107.

G. A. Leonov, N. V. Kuznetsov, M. A. Kiseleva, E. P. Solovyeva, and A. M. Zaretskiy, “Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor,” Nonlin. Dyn. 77, 277–288 (2014).CrossRef

108.

M. A. Kiseleva, N. V. Kondratyeva, N. V. Kuznetsov, G. A. Leonov, and E. P. Solovyeva, “Hidden periodic oscillations in drilling system driven by induction motor,” IFAC Proc. Vols. 47, 5872–5877 (2014).

109.

M. A. Kiseleva, N. V. Kondratyeva, N. V. Kuznetsov, and G. A. Leonov, “Hidden oscillations in drilling systems with salient pole synchronous motor,” IFAC-PapersOnLine 48, 700–705 (2015).CrossRef

110.

O. A. Ladyzhenskaya, “On finding minimal global attractors for the Navier–Stokes equations and other partial differential equations,” Usp. Mat. Nauk 42, 25–60 (1987).MathSciNet

111.

G. A. Leonov and V. Reitmann, Attraktoreingrenzung fur Nichtlineare Systeme (Teubner, Stuttgart, Leipzig, 1987).MATHCrossRef

112.

G. A. Leonov and N. V. Kuznetsov, “On differences and similarities in the analysis of Lorenz, Chen, and Lu systems,” Appl. Math. Comput. 256, 334–343 (2015).MathSciNetMATH

113.

D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Phys. Rep. 637, 1–50 (2016).MathSciNetMATHCrossRef

114.

N. V. Kuznetsov, G. A. Leonov, T. N. Mokaev, A. Prasad, and M. D. Shrimali, “Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system,” Nonlin. Dyn. 92, 267–285 (2018).MATHCrossRef

115.

G. Chen, N. V. Kuznetsov, G. A. Leonov, and T. N. Mokaev, “Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems,” Int. J. Bifurcat. Chaos Appl. Sci. Eng. 27, 1750115 (2017).MathSciNetMATHCrossRef

116.

G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Hidden attractor in smooth Chua systems,” Phys. D: Nonlin. Phenom. 241, 1482–1486 (2012).MathSciNetMATHCrossRef

117.

M.-F. Danca and N. V. Kuznetsov, “Hidden chaotic sets in a hopfield neural system,” Chaos, Solitons Fractals 103, 144–150 (2017).MathSciNetMATHCrossRef

118.

M.-F. Danca, N. Kuznetsov, and G. Chen, “Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system,” Nonlin. Dynam. 88, 791–805 (2017).MathSciNetCrossRef

119.

G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Commun. Nonlin. Sci. Numer. Simul. 28, 166–174 (2015).MathSciNetMATHCrossRef

120.

A. L. Fradkov and R. J. Evans, “Control of chaos: Methods and applications in engineering,” Ann. Rev. Control 29, 33–56 (2005).CrossRef

121.

A. L. Fradkov, Cybernetic Physics: Principles and Examples (Nauka, St. Petersburg, 2003) [in Russian].

122.

O. E. Rössler, “Continuous chaos – four prototype equations,” Ann. N. Y. Acad. Sci. 316, 376–392 (1979).MathSciNetMATHCrossRef

123.

N. V. Kuznetsov and T. N. Mokaev, “Numerical analysis of dynamical systems: Unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension,” J. Phys.: Conf. Ser. 1205, 012034 (2019).

124.

N. V. Kuznetsov, T. N. Mokaev, E. V. Kudryashova, O. A. Kuznetsova, and M.-F. Danca, “On lower-bound estimates of the Lyapunov dimension and topological entropy for the Rössler systems,” IFAC-PapersOnLine 52 (18), 97–102 (2019).CrossRef

125.

B. T. Polyak, “Stabilizing chaos with predictive control,” Autom. Remote Control 66, 1791 (2005).MathSciNetMATHCrossRef

126.

P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Control of multistability in hidden attractors,” Eur. Phys. J. Spec. Top. 224, 1485–1491 (2015).CrossRef

127.

P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Controlling dynamics of hidden attractors,” Int. J. Bifurcat. Chaos Appl. Sci. Eng. 25, 1550061 (2015).MathSciNetMATHCrossRef

128.

G. A. Leonov, M. M. Shumafov, and N. V. Kuznetsov, “Delayed feedback stabilization of unstable equilibria,” IFAC Proc. Vols. 19, 6818–6825 (2014).

129.

G. A. Leonov, M. M. Shumafov, and N. V. Kuznetsov, “Delayed feedback stabilization and the Huijberts-Michiels-Nijmeijer problem,” Differ. Equat. 52, 1707–1731 (2016).MathSciNetMATHCrossRef

130.

G. A. Leonov, I. Adan, B. R. Andrievsky, N. V. Kuznetsov, and A. Yu. Pogromsky, “Nonlinear problems in control of manufacturing systems,” IFAC Proc. Vols. 46 (9), 33–42 (2013).

131.

B. R. Andrievskii, N. V. Kuznetsov, G. A. Leonov, and A. Yu. Pogromskii, “Observed-based distributed control of production line,” Mekhatron. Avtomatiz. Upravl., No. 5, 13–25 (2014).

132.

A. L. Fradkov, Network Control Issues (IKI, Moscow, Izhevsk, 2015) [in Russian].

133.

D. Dudkowski, N. V. Kuznetsov, and T. N. Mokaev, “Chimera states and hidden attractors,” Phys. Life Rev. 28, 131–133 (2019).CrossRef

134.

N. V. Kuznetsov, “The Lyapunov dimension and its estimation via the Leonov method,” Phys. Lett. A 380, 2142–2149 (2016).MathSciNetMATHCrossRef

135.

N. V. Kuznetsov, T. A. Alexeeva, and G. A. Leonov, “Invariance of Lyapunov exponents and lyapunov dimension for regular and irregular linearizations,” Nonlin. Dyn. 85, 195–201 (2016).MathSciNetMATHCrossRef

136.

N. V. Kuznetsov and V. Reitmann, Attractor Dimension Estimates for Dynamical Systems: Theory and Computation (Dedicated to Gennady Leonov) (Springer, Cham, 2021).

137.

A. S. Matveev and A. Yu. Pogromsky, “Observation of nonlinear systems via finite capacity channels. pt. II. Restoration entropy and its estimates,” Automatica 103, 189–199 (2019).MATHCrossRef

138.

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139.

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140.

N. N. Bautin, “On the number of limit cycles that appear when coefficients change from an equilibrium state such as a focus or center,” Mat. Sb. 30 (72), 181–196 (1952).

141.

N. V. Kuznetsov and G. A. Leonov, “Lyapunov quantities, limit cycles and strange behavior of trajectories in two-dimensional quadratic systems,” J. Vibroeng. 10, 460–467 (2008).

142.

G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova, “A direct method for calculating Lyapunov quantities of two-dimensional dynamical systems,” Proc. of the Steklov Institute of Math. (Tr. Inst. Mat. Mekh. UrO RAN), 272 (SUPPL 1), 119–127 (2011).

143.

G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova, “Local methods for dynamic system cycle investigation,” Plenary Lecture on Pyatnitskiy 10th International Workshop on Stability and Oscillations of Nonlinear Control Systems, Moscow, 2008.

144.

V. I. Arnold, Experimental Mathematics (Moscow, FAZIS, 2005).

145.

N. V. Kuznetsov, O. A. Kuznetsova, and G. A. Leonov, “Visualization of four normal size limit cycles in two-dimensional polynomial quadratic system,” Differ. Equat. Dyn. Syst. 21, 29–34 (2013).MathSciNetMATHCrossRef

146.

I. G. Petrovskii and E. M. Landis, “On the number of limit cycles of the equation dy/dx = P(x, y)/Q(x, y), where P and Q are polynomials of the second degree,” Mat. Sb. 37 (79), 209–250 (1955).MathSciNet

147.

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148.

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149.

L. N. Belyustina, V. V. Bykov, K. G. Kiveleva, and V. D. Shalfeev, “On the size of the PLL locking system with a proportional-integrating filter,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 13, 561–567 (1970).

150.

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151.

V. V. Matrosov and V. D. Shalfeev, Nonlinear Dynamics of Phase Synchronization Systems (Nizhegor. Univ., Nizh. Novgorod, 2013) [in Russian].

152.

K. D. Aleksandrov, N. V. Kuznetsov, G. A. Leonov, N. Neittaanmaki, M. V. Yuldashev, and R. V. Yuldashev, “Computation of the lock-in ranges of phase-locked loops with PI filter,” IFAC-PapersOnLine 49 (14), 36–41 (2016).CrossRef

153.

M. V. Kapranov, “Capture phase locked loop,” Radiotekhnika 11 (12), 37–52 (1956).

154.

R. E. Best, N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Nonlinear analysis of phase-locked loop based circuits,” in Discontinuity and Complexity in Nonlinear Physical Systems (Springer, Cham, 2014), Vol. 6, pp. 169–192.MATH

155.

E. V. Kudryashova, N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Nonlinear analysis of PLL by the harmonic balance method: Limitations of the pull-in range estimation,” IFAC-PapersOnLine 50, 1451–1456 (2017).CrossRef

156.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Hold-in, pull-in, and lock-in ranges of PLL circuits: Rigorous mathematical definitions and limitations of classical theory,” IEEE Trans. Circuits Syst. I: Regular Papers 62, 2454–2464 (2015).MathSciNetCrossRef

157.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE,” Commun. Nonlin. Sci. Numer. Simul. 51, 39–49 (2017).CrossRef

158.

G. Bianchi, N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Limitations of PLL simulation: Hidden oscillations in MATLAB and SPICE,” in Proceedings of the International Congress on Ultra Modern Telecommunications and Control Systems and Workshops ICUMT,2015, Brno, 2016, pp. 79–84.

159.

G. Bianchi, N. V. Kuznetsov, G. A. Leonov, S. M. Seledzhi, M. V. Yuldashev, and R. V. Yuldashev, “Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear VCO,” IFAC-PapersOnLine 49 (14), 45–50 (2016).CrossRef

160.

S. J. Goldman, Phase-Locked Loops Engineering Handbook for Integrated Circuits (Artech House, Boston, 2007).

161.

G. A. Leonov, V. Reitmann, and V. B. Smirnova, Nonlocal Methods for Pendulum-like Feedback Systems (Teubner, Stuttgart, Leipzig, 1992).MATHCrossRef

162.

G. A. Leonov, I. M. Burkin, and A. I. Shepelyavy, Frequency Methods in Oscillation Theory (Kluwer, Dordretch, 1996).CrossRef

163.

G. A. Leonov and S. M. Seledzhi, Phase Synchronization Systems in Analog and Digital Circuitry (Nevskii Dialekt, St. Petersburg, 2002) [in Russian].

164.

G. A. Leonov, N. V. Kuznetsov, and S. M. Seledzhi, “Automation control – theory and practice,” in Nonlinear Analysis and Design of Phase-Locked Loops (InTech, London, 2009), pp. 89–114.CrossRef

165.

G. A. Leonov and N. V. Kuznetsov, Nonlinear Mathematical Models of Phase-Locked Loops. Stability and Oscillations (Cambridge Sci., Cottenham, 2014).

166.

N. V. Kuznetsov, “The development of stability theory for differential inclusions in G. A. Leonova and its application for stability analysis of angular orientation systems,” in Proceedings of the Korolev’s Readings (Moscow, 2019).

167.

N. V. Kuznetsov, M. Y. Lobachev, M. V. Yuldashev, R. V. Yuldashev, E. V. Kudryashova, O. A. Kuznetsova, E. N. Rosenwasser, and S. M. Abramovich, The birth of the global stability theory and the theory of hidden oscillations. In Proc. of European Control Conf. St. Petersburg, P. 769–774 (2020).

168.

D. Abramovitch, “Analysis and design of a third order phase-lock loop,” in Proceedings of the Conference on 21st Century Military Communications—What’s Possible?, San Diego,1988, pp. 455–459.

169.

S. Yu. Zheltov, K. K. Veremeenko, N. V. Kim, D. A. Kozorez, M. N. Krasilshchikov, G. G. Sebryakov, K. I. Sypalo, and A. I. Chernomorskiy, Modern information technologies in the tasks of navigation and guidance of unmanned maneuverable aerial vehicles (Fizmatlit, Moscow, 2009) [in Russian].

170.

E. D. Kaplan and C. J. Hegarty, Understanding GPS: Principles and Applications (Artech House, Boston, 2006).

171.

I. A. Kalyaev, I. I. Levin, E. A. Semernikov, and V. I. Shmoilov, Reconfigurable Multi-Pipeline Computing Structures (YuNTs RAN, Rostov-on-Don, 2008) [in Russian].

172.

S. I. Vol’skii, N. V. Kuznetsov, D. A. Sorokin, M. V. Yuldashev, and R. V. Yuldashev, “The choice of the frequency of the output voltage of the magnetoelectric generator of the power supply system of aircraft with electric traction,” in Proceedings of the International Conference on Electrical Complexes and Systems, Ufa,2020. http://www.icoecs.com/.

173.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Analytical method for computation of phase-detector characteristic,” IEEE Trans. Circuits Syst. II: Express Briefs 59, 633–647 (2012).CrossRef

174.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large,” Signal Process. 108, 124–135 (2015).CrossRef

175.

R. E. Best, N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Simulation of analog Costas loop circuits,” Int. J. Autom. Comput. 11, 571–579 (2014).MATHCrossRef

176.

R. E. Best, N. V. Kuznetsov, O. A. Kuznetsova, et al., “A short survey on nonlinear models of the classic Costas loop: Rigorous derivation and limitations of the classic analysis,” in Proceedings of the American Control Conference (IEEE, Chicago, 2015), Art. No. 7170912, pp. 1296–1302.

177.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Computation of the phase detector characteristic of classical PLL,” Dokl. Math. 91, 246–249 (2015).MathSciNetMATHCrossRef

178.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Mathematical models of the Costas loop,” Dokl. Math. 92, 594–598 (2015).MathSciNetMATHCrossRef

179.

G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, and R. V. Yuldashev, “Computation of the phase detector characteristic of a QPSK Costas loop,” Dokl. Math. 93, 348–353 (2016).MathSciNetMATHCrossRef

180.

N. V. Kuznetsov, M. V. Yuldashev, R. V. Yuldashev, M. V. Blagov, E. V. Kudryashova, O. A. Kuznetsova, and T. N. Mokaev, “Comments on Van Paemel’s mathematical model of charge-pump phase-locked loop,” Differ. Uravn. Protsessy Upravl. 1, 109–120 (2019). https://diffjournal.spbu.ru/pdf/19107-jdecp-kuznetsov.pdfMathSciNetMATH

181.

R. E. Best, N. V. Kuznetsov, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “Tutorial on dynamic analysis of the Costas loop,” IFAC Ann. Rev. Control 42, 27–49 (2016).CrossRef

182.

N. V. Kuznetsov, O. A. Kuznetsova, G. A. Leonov, M. V. Yuldashev, and R. V. Yuldashev, “A short survey on nonlinear models of QPSK Costas loop,” IFAC-PapersOnLine 50, 6525–6533 (2017).CrossRef

183.

N. V. Kuznetsov, M. Yu. Lobachev, M. V. Yuldashev, and R. V. Yuldashev, “On the Gardner problem for the phase-locked loops,” Dokl. Math. 100, 568–570 (2019).MATHCrossRef

184.

G. A. Leonov and N. V. Kuznetsov, “Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems,” IFAC Proc. Vols. 44, 2494–2505 (2011).

185.

N. V. Kuznetsov, “Plenary lecture “Hidden attractors in fundamental problems and engineering models. A short survey”,” Lect. Notes Electric. Eng. 371, 13–25 (2016).

186.

N. V. Kuznetsov, “Plenary lecture "Theory of hidden oscillations and stability of control systems,” in Proceedings of the International Conference on Stability, Control, Differential Games, Devoted to the 95th Anniversary of Academician N. N. Krasovskii (Yekaterinburg, 2019), pp. 201–204.

187.

N. V. Kuznetsov, “Theory of hidden oscillations and the stability of control systems,” in Proceedings of the 12th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics (Ufa, 2019), Vol. 1, pp. 46–48. www.youtube.com/watch?v=843m-rI5nTM.

188.

N. V. Kuznetsov, “Control of oscillations in phase locked loop systems,” in Proceedings of the 12th Multiconference on Control Problems, Divnomorskoe,2019, Vol. 3, pp. 21–26.

Titel: Theory of Hidden Oscillations and Stability of Control Systems
verfasst von: N. V. Kuznetsov
Publikationsdatum: 01.09.2020
Verlag: Pleiades Publishing
Erschienen in: Journal of Computer and Systems Sciences International / Ausgabe 5/2020
Print ISSN: 1064-2307
Elektronische ISSN: 1555-6530
DOI: https://doi.org/10.1134/S1064230720050093

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