## Application of the Harmonic Linearization Method to the Study a Control Systems with a Self-Oscillatory Regime

**Authors:**Kulyabov D.S., Korolkova A.V., Velieva T.R.**Issue:**Vol 25, No 3 (2017)**Pages:**234-252**URL:**http://journals.rudn.ru/miph/article/view/16206**DOI:**http://dx.doi.org/10.22363/2312-9735-2017-25-3-234-252

#### Abstract

In data transmission networks implemented as systems with control, the phenomenon of global synchronization can occur. Outwardly, this phenomenon manifests itself as a self-oscillating mode in the system. This mode negatively affects the characteristics of the entire system, such as throughput and transmission delays. Relevant is the problem of finding the areas of occurrence of self-oscillation. The authors investigated this problem for the system as a whole. Also, the problem of isolating the elements of the system responsible for the appearance of an autooscillatory regime is urgent. The complexity of this problem is caused by the essentially nonlinear character of the system and its elements. Often, the linearization method is used for the decomposition of the system. But with the linearization, the self-oscillatory regime disappears. There is a need to find a method of decomposition, non-destructive self-oscillating mode of the system. As such a method, the authors suggest using the method of harmonic linearization. This method is used in the control theory. However, we must admit that this mathematical apparatus is little known to researchers specializing in the study of networks. The authors tried to describe in as much detail the process of research using the method of harmonic linearization. The method is used to study the influence of the form of RED-type function on the occurrence of self-oscillation mode. Thus, this material is more methodical than exploratory one.

### D S Kulyabov

kulyabov_ds@rudn.university

<p>Peoples’ Friendship University of Russia (RUDN University</p>

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Department of Applied Probability and Informatics; Laboratory of Information Technologies Joint Institute for Nuclear Research 6 Joliot-Curie St., Dubna, Moscow region, 141980, Russian Federation

### A V Korolkova

korolkova_av@rudn.university

<p>Peoples’ Friendship University of Russia (RUDN University</p>

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Department of Applied Probability and Informatics

### T R Velieva

velieva_tr@rudn.university

<p>Peoples’ Friendship University of Russia (RUDN University</p>

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Department of Applied Probability and Informatics

- S. Floyd, V. Jacobson, Random Early Detection Gateways for Congestion Avoidance, IEEE/ACM Transactions on Networking 1 (4) (1993) 397–413. doi: 10.1109/90.251892.
- V. Jacobson, Congestion Avoidance and Control, ACM SIGCOMM Computer Communication Review 18 (4) (1988) 314–329. arXiv:arXiv:1011.1669v3, doi: 10.1145/52325.52356.
- A.V. Korolkova, D.S. Kulyabov, A.I. Chernoivanov, On the Classification of RED Algorithms, Bulletin of Peoples’ Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (3) (2009) 34–46, in Russian.
- A.V. Korolkova, D.S. Kulyabov, Mathematical Model of the Dynamic Behavior of RED-Like System Parameters, Bulletin of Peoples’ Friendship University of Russia. Series “Mathematics. Information Sciences. Physics” (1) (2010) 54–64, in Russian.
- T.R. Velieva, A.V. Korolkova, D.S. Kulyabov, B.A. Dos Santos, Model Queue Management on Routers, Bulletin of Peoples’ Friendship University of Russia. Series “Mathematics. Information Sciences. Physics” 2 (2014) 81–92, in Russian.
- T.R. Velieva, A.V. Korolkova, D.S. Kulyabov, Designing Installations for Verification of the Model of Active Queue Management Discipline RED in the GNS3, in: 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), IEEE Computer Society, 2015, pp. 570–577. arXiv:1504.02324, doi: 10.1109/ICUMT.2014.7002164.
- K.J. Aström, R.M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, 2008.
- H. Nyquist, Regeneration Theory, Bell System Technical Journal 11 (1) (1932) 126– 147. doi: 10.1002/j.1538-7305.1932.tb02344.x.
- J. Hsu, A. Meyer, Modern Control Principles and Applications, McGraw-Hill, 1968.
- A.A. Voronov, D.P. Kim, V.M. Lokhin, et al., Theory of Automatic Control: Textbook for High Schools on the Specialty “Automation and telemechanics”. In two parts. Part II. Theory of Nonlinear and Special Systems of Automatic Control, 2nd Edition, High School, Moscow, 1986, in Russian.
- E.J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, Macmillan, 1877.
- A. Hurwitz, Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Mathematische Annalen 46 (2) (1895) 273–284. doi: 10.1007/BF01446812.
- F.R. Gantmacher, The Theory of Matrices, Chelsea Pub. Co., 1959.
- V.A. Besekerskiy, E. P. Popov, Theory of Automatic Control Systems, Nauka, Moscow, 1972, in Russian.
- N.A. Babakov, A.A. Voronov, A.A. Voronova, et al., Theory of Automatic Control: Textbook for High Schools on the Specialty “Automation and telemechanics”. In two parts. Part. I. Theory of Linear Systems of Automatic Control, 2nd Edition, High School, Moscow, 1986, in Russian.
- M. Allman, V. Paxson, E. Blanton, TCP Congestion Control, Tech. rep. (sep 2009). doi: 10.17487/rfc5681.
- V. Kushwaha, R. Gupta, Congestion Control for High-Speed Wired Network: A Systematic Literature Review, Journal of Network and Computer Applications 45 (2014) 62–78. doi: 10.1016/j.jnca.2014.07.005.
- R. Adams, Active Queue Management: A Survey, IEEE Communications Surveys Tutorials 15 (3) (2013) 1425–1476. doi: 10.1109/SURV.2012.082212.00018.
- A. Jenkins, Self-Oscillation, Physics Reports 525 (2) (2013) 167–222. arXiv:1109.6640, doi: 10.1016/j.physrep.2012.10.007.
- F. Ren, C. Lin, B. Wei, A Nonlinear Control Theoretic Analysis to TCP-RED System, Computer Networks 49 (4) (2005) 580–592. doi: 10.1016/j.comnet.2005.01.016.
- W. Lautenschlaeger, A. Francini, Global Synchronization Protection for Bandwidth Sharing TCP Flows in High-Speed Links, in: Proc. 16-th International Conference on High Performance Switching and Routing, IEEE HPSR 2015, Budapest, Hungary, 2015. arXiv:1602.05333.
- V. Misra, W.-B. Gong, D. Towsley, Stochastic Differential Equation Modeling and Analysis of TCP-Windowsize Behavior, Proceedings of PERFORMANCE 99.
- V. Misra, W.-B. Gong, D. Towsley, Fluid-Based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED, ACM SIGCOMM Computer Communication Review 30 (4) (2000) 151–160. doi: 10.1145/347057.347421.
- C. V. V. Hollot, V. Misra, D. Towsley, Wei-Bo Gong, On Designing Improved Controllers for AQM Routers Supporting TCP Flows, in: Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213), Vol. 3, IEEE, 2001, pp. 1726–1734. doi: 10.1109/INFCOM.2001.916670.
- C.V.V. Hollot, V. Misra, D. Towsley, A Control Theoretic Analysis of RED, in: Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213), Vol. 3, IEEE, 2001, pp. 1510–1519. doi: 10.1109/INFCOM.2001.916647.
- A.V. Korolkova, D.S. Kulyabov, L.A. Sevastianov, Combinatorial and Operator Approaches to RED Modeling, Mathematical Modelling and Geometry 3 (3) (2015) 1–18.
- A.V. Korolkova, T.R. Velieva, P.A. Abaev, L.A. Sevastianov, D.S. Kulyabov, Hybrid Simulation of Active Traffic Management, Proceedings 30th European Conference on Modelling and Simulation (2016) 685–691doi: 10.7148/2016-0685.
- R. Brockett, Stochastic Analysis for Fluid Queueing Systems, in: Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), Vol. 3, IEEE, 1999, pp. 3077–3082. doi: 10.1109/CDC.1999.831407.
- E. Altman, T. Jim´enez, NS Simulator for Beginners, Synthesis Lectures on Communication Networks 5 (1) (2012) 1–184. doi: 10.2200/S00397ED1V01Y201112CNT010.
- T. Issariyakul, E. Hossain, Introduction to Network Simulator NS2, Vol. 9781461414, 2012. doi: 10.1007/978-1-4614-1406-3.

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