Анализ колебаний в динамических системах и приложения к системам управления: обзор современных исследований в СПбГУ и ТУ Шарифа и будущие перспективы [Oscillatory Behavior Analysis in Dynamical Systems and its Applications in Control Systems: An Overview on Recent Advances at SPbU and SUT and Future Perspective]

Project

Project Details

Description

a) Analysis of regular/irregular oscillations generated by oscillatory systems, due to the interdisciplinary nature and diverse applicability of such dynamic systems, has been a hot research topic in recent years. Active research groups at SPbU and SUT parallelly work on this topic.
The research group with the leadership of Prof. Kuznetsov at SPbU is working on the development of effective analytical-numerical methods for the reliable analysis of oscillations in dynamical systems and their applications to various fundamental problems and applied dynamical models in automatic control, electronics, physics and other fields. The theory of oscillations, developed by A.A. Andronov and his followers in the first half of 20th century, made it possible to effectively analyze periodic attractors in low dimensional dynamical models. Then, in the second half of 20th century, various self-exciting chaotic attractors were discovered and the theory of birth of such attractors from unstable equilibria was well developed. Recent discovery of hidden chaotic attractors at the beginning of 21th century, a necessity to analyze multistable multidimensional dynamic systems, as well as the emergence of the modern computing tools shows the relevance and necessity of the modernization of Andronov's theory, and the development of the theory of hidden oscillations [1-4]. The climate, a number of ecosystems, the human brain, arrays of coupled lasers, financial markets and many applied engineering systems are modeled by complex dynamical systems which are characterized by the existence of many coexisting attractors. This property of the systems is called multistability and refers to systems that are neither stable nor totally unstable, but that alternate between two or more mutually exclusive states (attractors) over time over time and under external disturbances. Multistable systems are very sensitive towards noise, initial conditions and system parameters. In multistable systems, particularly in the case of the existence of hidden attractors with very small basins or previously unidentified attractors, one can observe the sudden switch to unexpected (undesired or unknown) attractors. Such a shift can lead to the catastrophic events ranging from sudden climate changes, serious diseases to financial crises and disasters of commercial devices. To keep the system on the desired attractor (trivial - stable equilibrium point, or oscillating attractor) one needs first to uncover all coexisting attractors and next apply an appropriate controlling scheme. One of the areas for application of the methods developed in the Prof. Kuznetsov's group is the study of dynamical models described by the fractional-order differential equations and maps. The range of problems for the fractional-order systems includes the study of the complex limiting oscillations [5,6], the localization of hidden oscillations and attractors [7], their characterization [8,9] and control of chaos [10].
The main works of the research group with the leadership of Tavazoei at SUT have focused on fractional order oscillatory systems and its applications in control system design. Analysis of chaotic oscillations in fractional order systems [11], providing a rigorous proof for absence of exactly periodic solutions in fractional order systems [12, 13], design of multi-frequency fractional oscillators [14, 15], dynamical behavior analysis of uncertain Lotka-Volterra systems [16-18], analysis of oscillations in relay feedback systems containing fractional order sub-systems [19], harmonic analysis in fractional order oscillatory systems [20], analytically finding the oscillatory region in order space for nonlinear fractional-order systems [21], proposing effective methods for suppression of undesirable oscillations [22], and study on the influence of the fractionality nature of the elements in oscillations generated by electrical networks [23] are some relevant samples of the research works done by this group.
Considering the proximity of the research interests of the above-mentioned research groups at SPbU and SUT and in order to foster the synergies between them, this joint project aims to provide the basic foundation for establishment of a long-term cooperation between the aforementioned research groups. By doing this joint project, overview reports on the studies done by SPbU and SUT researchers on the subject of oscillatory behavior analysis in dynamical systems and their practical application in control systems design will be provided. Also, a comprehensive joint research program, including the list and definitions of the joint projects on "oscillatory behavior analysis in dynamical systems" that can be done by the joint research team of SPbU and SUT researchers in the long term, will be proposed. Furthermore, a sample project from the proposed list of joint projects will be done.
In this project classical methods for analyzing the stability and oscillations as well as the methods developed in the Russian and Iranian groups, presented in highly rated publications [1-23], will be used. Also we will apply and develop methods based on artificial intelligence to predict and study the complex dynamics of systems with fractional derivatives. For this, he will use the cooperation of the Russian team: with the group of Prof. Zelinkа (Czech Republic) - a well-known specialist in evaluation algorithms; and with the group of Prof. P. Neittaanmäki (Finland, honorary professor of St. Petersburg State University) - a well-known specialist in computing and information technology. Remark that reliable scientific computing are considered strategic not only by many research groups around the world but also included in the strategic research plan of in many countries (including Russian and Iran). With reliable computing and simulation, we expect to have immediate industrial and societal impact with direct applications to every-day-life problems.
The interdisciplinary nature of the approaches, proposed for the development in the project, is confirmed by the inclusion of the head of the Russian group in the list of WoS Highly Cited Researchers in the field cross-field category in 2019, and by publications of the Iranian group’s leader Mohammad Saleh Tavazoei in top journals of various applied fields (e.g. Automatica, IEEE Transactions on Circuits and Systems I, IEEE Transactions on Signal Processing, IEEE Transactions on Automatic Control, IEEE Transactions on Industrial Electronics and others).

References
[1] G.A. Leonov, N.V. Kuznetsov, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits”, International Journal of Bifurcation and Chaos, 23(1), 2013, art. no. 1330002 [Q1, www.scimagojr.com]
[2] G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion”, European Physical Journal Special Topics, 224, 2015, pp. 1421-1458 [Q2, www.scimagojr.com]
[3] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, A. Prasad, “Hidden attractors in dynamical systems”, Physics Reports, 637, 2016, pp. 1-50 [Q1, www.scimagojr.com]
[4] N.V. Kuznetsov, G.A. Leonov, T.N. Mokaev, A. Prasad, M.D. Shrimali, “Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system”, Nonlinear Dynamics, 92(2), 2018, 267-285 [Q1, www.scimagojr.com]
[5] G.M. Mahmoud, A.A. Farghaly, A.H. Shoreh, “A technique for studying a class of fractional-order nonlinear dynamical systems”. International Journal of Bifurcation and Chaos, 27(09), 2017, p.1750144.
[6] A.A. Farghaly, A.H. Shoreh, “Some complex dynamical behaviors of the new 6D fractional-order hyperchaotic Lorenz-like system”. Journal of the Egyptian Mathematical Society, 26(1), 2018, pp.138-155.
[7] M.F. Danca, M. Feckan, N.V. Kuznetsov, G. Chen, “Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system”. Nonlinear Dynamics, 91(4), 2018, pp.2523-2540. [Q1, www.scimagojr.com]
[8] M.F. Danca, M. Feckan, N.V. Kuznetsov, G.Chen, “Fractional-order PWC systems without zero Lyapunov exponents”. Nonlinear Dynamics, 92(3), 2018, pp.1061-1078. [Q1, www.scimagojr.com]
[9] M.F. Danca, N.V. Kuznetsov, “Matlab code for Lyapunov exponents of fractional-order systems”. International Journal of Bifurcation and Chaos, 28(05), 2018, p.1850067. [Q1, www.scimagojr.com]
[10] M.F. Danca, M. Feckan, N.V. Kuznetsov, “Chaos control in the fractional order logistic map via impulses”. Nonlinear Dynamics, 98(2), 2019, pp.1219-1230. [Q1, www.scimagojr.com]
[11] M. S. Tavazoei, "Fractional order chaotic systems: history, achievements, applications, and future challenges," The European Physical Journal Special Topics, Volume 229, 2020, Pages 887-904.
[12] M. S. Tavazoei and M. Haeri, “A Proof for non Existence of Periodic Solutions in Time Invariant Fractional Order Systems,” Automatica, Volume 45, Issue 8, August 2009, Pages 1886-1890.
[13] M. S. Tavazoei, "A Note on Fractional-Order Derivatives of Periodic Functions," Automatica, Volume 46, Issue 5, May 2010, Pages 945-948. [Q1, www.scimagojr.com]
[14] M. S. Tavazoei, "Upper and Lower Bounds for the Maximum Number of Frequencies That Can Be Generated by a Class of Fractional Oscillators," IEEE Transactions on Circuits and Systems I, Volume 66, Issue 4, 2018, Pages 1584-1593. [Q1, www.scimagojr.com]
[15] M. S. Tavazoei, M. Haeri, M. Siami and S. Bolouki, "Maximum Number of Frequencies in Oscillations Generated by Fractional Order LTI Systems," IEEE Transactions on Signal Processing, Volume 58, Issue 8, August 2010, Pages 4003-4012 [Q1, www.scimagojr.com]
[16] V. Badri, M. S. Tavazoei, and M. J. Yazdanpanah, “Global Stabilization of Uncertain Lotka-Volterra Systems via Positive Nonlinear State Feedback,” IEEE Transactions on Automatic Control, In Press, 2020. DOI: 10.1109/TAC.2020.2972832 [Q1, www.scimagojr.com]
[17] V. Badri, M. J. Yazdanpanah, and M. S. Tavazoei, "Global Stabilization of Lotka–Volterra Systems With Interval Uncertainty," IEEE Transactions on Automatic Control, Volume 64, Issue 3, 2018, Pages 1209-1213. [Q1, www.scimagojr.com]
[18] V. Badri, M. J. Yazdanpanah, and M. S. Tavazoei, "On Stability and Trajectory Boundedness of Lotka–Volterra Systems with Polytopic Uncertainty," IEEE Transactions on Automatic Control, Volume 62, Issue 12, 2017, Pages 6423-6429. [Q1, www.scimagojr.com]
[19] D. Rezaei and M. S. Tavazoei, "Analysis of Oscillations in Relay Feedback Systems with Fractional-Order Integrating Plants," ASME Journal of Computational and Nonlinear Dynamics, Volume 12, Issue 5, 2017, 051023.
[20] M. Siami and M. S. Tavazoei, "Oscillations in fractional order LTI Systems: Harmonic Analysis and Further Results," Signal Processing, Volume 93, Issue 5, May 2013, Pages 1243-1250. [Q1, www.scimagojr.com]
[21] M. S. Tavazoei, "Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems," ASME Journal of Computational and Nonlinear Dynamics, Volume 9, Issue 2, 2014, 021011.
[22] M. S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, and M. Siami, “Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations,” IEEE Transactions on Industrial Electronics, Volume 55, Number 11, November 2008, Pages 4094-4101. [Q1, www.scimagojr.com]
[23] M. Tavakoli-Kakhki and M. S. Tavazoei, "Static Feedback versus Fractionality of the Electrical Elements in the Van der Pol Circuit," Nonlinear Dynamics, Volume 72, Issues 1-2, April 2013, Pages 365-375. [Q1, www.scimagojr.com]

b) To expand funding opportunities and ensure the continuity of the proposed project, it is proposed to involve a group from Amirkabir University of Technology headed by Dr. Sajad Jafari (WoS Highly Cited Researcher 2019), with whom the Russian team already has highly cited joint publications in high-ranking journals, and apply for a joint Russian-Iranian grant from the Russian Foundation of Basic Research this year (heads of Dr. Sajad Jafari and N.V. Kuznetsov). After joint papers within the current project with Dr. Mohammad Saleh Tavazoei will be published, we are going to apply for a joint Russian-Iranian grant from the Russian Foundation of Basic Research (the leaders of Dr. Mohammad Saleh Tavazoei and N.V. Kuznetsov).

c) Possible results of intellectual activity that may arise during the trip are the following: joint publications, materials of joint seminars for graduate students and young researchers.
AcronymJFS SUT 2020
StatusActive
Effective start/end date21/10/2018/12/20