Analysis of interconnected oscillators by dissipativity theory

IEEE Transactions on Automatic Control

Volumn 52, Issue 2, 2007, Pages 256-270

  Stan, Guy Bart ^a   Sepulchre, Rodolphe ^b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

Global limit cycle analysis; Global synchronization; Hopf and pitchfork bifurcations; Networks of oscillators

Indexed keywords

DYNAMICAL SYSTEMS; HOPF BIFURCATION; STABILITY; SYNCHRONIZATION;

DISSIPATIVITY THEORY; GLOBAL LIMIT CYCLE ANALYSIS; INTERCONNECTED OSCILLATORS; PITCHFORK BIFURCATIONS;

OSCILLATORS (ELECTRONIC);

EID: 33947407698     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2006.890471     Document Type: Article

References (42)

1. A. Winfree, The Geometry of Biological Time, 2nd ed. New York: Springer-Verlag, 2000.
2. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge, U.K.: Cambridge Univ. Press, 1996.
3. S. H. Strogatz, Nonlinear Dynamics and Chaos. Cambridge, MA: Westview Press, 2000.
4. S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order. New York: Hyperion, 2003.
5. J. Willems, "Dissipative dynamical systems," Arch. Rat. Mech. Anal., vol. 45, pp. 321-393, 1972.
6. 2-Gain and Passivity Techniques in Nonlinear Control. London, U.K.: Springer-Verlag, 2000.
7. R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. London, U.K.: Springer-Verlag, 1997.
8. L. O. Chua, "Passivity and complexity," IEEE Trans. Circuits Syst. I, vol. 46, no. 1, pp. 71-82, Jan. 1999.
9. R. Sepulchre and G.-B. Stan, "Feedback mechanisms for global oscillations in Lure systems," Syst. Control Lett., vol. 54, no. 8, pp. 809-818, 2005.
10. H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
11. H. R. Wilson, Spikes, Decisions and Actions - Dynamical Foundations of Neuroscience. New York: Oxford Univ. Press, 1999.
12. V. A. Yakubovich, "Frequency-domain criteria for oscillation in nonlinear systems with one stationary nonlinear component," Siberian Math. J., vol. 15, no. 5, pp. 1100-1129, 1973.
13. E. A. Tomberg and V. A. Yakubovich, "Conditions for auto-oscillations in nonlinear systems," Siberian Math. J., vol. 30, no. 4, pp. 180-194, 1989.
14. G. Leonov, I. Burkin, and A. Shepeljavyi, Frequency Methods in Oscillation Theory. Norwell, MA: Kluwer, 1996.
15. A. Mees, Dynamics of Feedback Systems. New York: Wiley-Interscience, 1981.
16. J. Goncalves, A. Megretski, and M. Dahleh, "Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions," IEEE Trans. Autom. Control, vol. 48, no. 12, pp. 2089-2106, Dec. 2003.
17. G.-B. Stan and R. Sepulchre, "Dissipativity and global analysis of limit cycles in networks of oscillators," in Proc. 16th Int. Symp. Mathematical Theory of Networks and Systems, Leuven, Belgium, 2004.
18. G.-B. Stan, "Global analysis and synthesis of oscillations: A dissipativity approach" Ph.D. dissertation, Univ. Liège, Liège, Belgium, Mar. 2005 [Online]. Available: http://www.montefiore.ulg.ac.be/stan
19. A. Graham, Kronecker Products and Matrix Calculus with Applications. New York: Ellis Horwood, 1981.
20. M. Arcak and A. Teel, "Input-to-state stability for a class of Lurie systems," Automatica, vol. 38, no. 11, pp. 1945-1949, 2002.
21. J. C. Willems, "Dissipative dynamical systems: Parts I and II," Arch. Rat. Mech. Anal., vol. 45, pp. 321-393, 1972.
22. C. I. Byrnes, A. Isidori, and J. C. Willems, "Passivity, feedback equivalence, and global stabilization of minimum phase systems," IEEE Trans. Autom. Control, vol. 36, no. 1, pp. 1128-1240, Nov. 1991.
23. G. Zames and P. L. Falb, "Stability conditions for systems with monotone and slope-restricted nonlinearities," SIAM J. Control Optim., vol. 6, no. 1, pp. 89-108, 1968.
24. M. G. Safonov and V. V. Kulkarni, "Zames-Falb multipliers for MIMO nonlinearities," Int. J. Robust Nonlinear Control, vol. 10, pp. 1025-1038, 2000.
25. A. Megretski and A. Rantzer, "System analysis via integral quadratic constraints," IEEE Trans. on Automatic Control, vol. 42, no. 6, pp. 819-830, Jun. 1997.
26. V. M. Popov, "Absolute stability of nonlinear control systems of automatic control," Automation and Remote Control, vol. 22, pp. 857-875, 1962.
27. H. K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996.
28. L. Moreau and D. Aeyels, "Practical stability and stabilization," IEEE Trans. Autom. Control, vol. 45, no. 8, pp. 1554-1558, Aug. 2000.
29. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, 1990, Texts in Applied Mathematics, 2.
30. J. Carr, Applications of Center Manifold Theory. New York: Springer-Verlag, 1981.
31. A. Isidori, Nonlinear Control Systems, 3rd ed. Berlin, Germany: Springer-Verlag, 1995.
32. J. D. Murray, Mathematical Biology, I: An Introduction, 3rd ed. New York: Springer-Verlag, 2002.
33. F. C. Hoppensteadt, An Introduction to the Mathematics of Neurons. Cambridge, U.K.: Cambridge Univ. Press, 1986.
34. D. Angeli, "A Lyapunov approach to incremental stability properties," IEEE Trans. Autom. Control, vol. 47, no. 3, pp. 410-422, Mar. 2002.
35. W. Lohmiller and J.-J. E. Slotine, "On contraction analysis for nonlinear systems," Automatica, vol. 34, no. 6, pp. 683-696, 1998.
36. A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer, "Convergent dynamics: A tribute to Boris Pavlovich Demidovich," Syst. Control Lett., vol. 52, pp. 257-261, 2004.
37. J. Willems, The Analysis of Feedback Systems. Cambridge, MA: MIT Press, 1971.
38. S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness. Upper Saddle River, NJ: Prentice-Hall, 1989.
39. J.-J. Slotine and W. Wang, "A study of synchronization and group cooperation using partial contraction theory," in Block Island Workshop on Cooperative Control, K. V., Ed. New York: Springer-Verlag, 2003.
40. A. Pogromsky, "Passivity based desing of synchronizing systems," Int. J. Bifur. Chaos, vol. 8, pp. 295-319, 1998.
41. L. B. Cremean and R. M. Murray, "Stability analysis of interconnected nonlinear systems under matrix feedback," in Proc. 42nd Conf. Decision Control, Maui, HI, Dec. 2003, vol. 4, pp. 3078-3083, (Part One).
42. R. Olfati-Saber and R. M. Murray, "Consensus problems in networks of agents with switching topology and time-delays," IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520-1533, Sep. 2004.