Abstract
Summary. This paper presents a feedback method to achieve synchronization of coupled identical oscillators which are characterized by polynomial vector fields. Here, synchronization means asymptotic coincidence of the states of all the systems. Even though their models are identical, the state trajectories of the identical systems are different because of different initial conditions. Unlike other approaches where just a linear damping term is added to each system in order to achieve synchronization, we design nonlinear coupling functions between the subsystems in such a way that stability of the error dynamics between any two models results. To do that, a certain dissipation inequality and sum of squares as a computational tool are used. Finally, two examples are presented to illustrate the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I.I. Blekhman, A.L. Fradkov, H. Nijmeijer, and A.Y. Pogromsky. On self-synchronization and controlled synchronization. Systems & Control Letters, 31:299–306, 1997.
R. Diestel. Graph Theory. Springer Verlag, 2005.
C. Ebenbauer. Polynomial Control Systems: Analysis and Design via Dissipation Inequalities and Sum of Squares. Ph.D dissertation, University of Stuttgart, Germany, 2005.
A. Isidori. Nonlinear Control Systems, Vol. II. Springer Verlag, 1999.
H.K. Khalil. Nonlinear Systems. Prentice Hall, 2002.
J. S. Kim and F. Allgöwer. A nonlinear synchronization scheme for multiple Hindmarsh-Rose models. Submitted for publication, 2007.
D. Mishra, A. Yadav, S. Ray, and P.K. Kalra. Controlling synchronization of modified Fitzhugh-Nagumo neurons under external electrical stimulation. NeuroQuantology, 4(1):50–67, 2006.
H. Nijmeijer and I.M.Y. Mareels. An observer looks at synchronization. IEEE Trans. on Circuits and Systems-I: Fundamental Theory and Applications, 44:882–890, 1997.
J.G. Ojalvo, M.B. Elowitz, and S. Strogatz. Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing. In Proc. Natl. Acad. Sci. U.S.A., volume 101, pages 10955–10960, 2004.
R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. on Automat. Contr., 49(9):1520–1533, 2004.
W. T. Oud and I. Tyukin. Sufficient conditions for synchronization of Hindmarsh and Rose neurons: passivity-based-approach. In Proc. of the 6th IFAC Symposium in Nonlinear Control Systems, Stuttgart, 2004.
P.A. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D dissertation, California Institute of Technology, 2000.
L. Pecora and M. Barahona. Synchronization of oscillators in complex networks. Chaos and Complexity Letters, 1:61–91, 2005.
R.D. Pinto, P. Varona, A.R. Volkovskii, A. Szucs, H.D.I. Abarbanel, and M.I. Rabinovich. Synchronous behavior of two coupled electronic neurons. Physical Review E, 62(2):2644–2656, 2000.
S. Prajna, A. Papachristodoulou, P. Seiler, and P.A. Parrilo. SOSTOOLS and its control applications, In Positive Polynomials in Control. Lecture Notes in Control and Information Sciences. Springer Verlag, 2005.
G.-B. Stan. Global Analysis and Synthesis of Oscillations: a Dissipativity Approach. Ph.D dissertation, University of Liege, Belgium, 2005.
G.-B. Stan, A.O. Hamadeh, J. Goncalves, and R. Sepulchre. Output synchronization in networks of cyclic biochemical oscillators. In Proc. of the 2007 Amer. Contr. Conf., New York, 2007.
S.H. Strogatz. From Kuramoto to crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143:1–20, 2000.
S.H. Strogatz. Exploring complex networks. Nature, 410:268–276, 2001.
Y. Wang, A.H. Guan, and H.O. Wang. Feedback and adaptive control for the synchronization of Chen system via a single variable. Physics Letters A, 312:34–40, 2003.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kim, JS., Allgöwer, F. (2008). Nonlinear Synchronization of Coupled Oscillators: The Polynomial Case. In: Astolfi, A., Marconi, L. (eds) Analysis and Design of Nonlinear Control Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74358-3_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-74358-3_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74357-6
Online ISBN: 978-3-540-74358-3
eBook Packages: EngineeringEngineering (R0)