A contraction theory approach to stochastic incremental stability

IEEE Transactions on Automatic Control

Volumn 54, Issue 4, 2009, Pages 816-820

  Pham, Quang Cuong ^a   Tabareau, Nicolas ^a   Slotine, Jean Jacques ^b  

^a CNRS   (France)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Incremental stability; Nonlinear contraction theory; Stochastic stability

Indexed keywords

CONTRACTION RATES; INCREMENTAL STABILITY; MEAN SQUARES; NOISE INTENSITIES; NON-LINEAR OBSERVERS; NONLINEAR CONTRACTION THEORY; STOCHASTIC DYNAMICAL SYSTEMS; STOCHASTIC STABILITY; STOCHASTIC SYNCHRONIZATIONS;

DYNAMICAL SYSTEMS; RANDOM PROCESSES; SHRINKAGE;

SYSTEM STABILITY;

EID: 67349182327     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2008.2009619     Document Type: Article

References (32)

1. J.-J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.
2. D. Lewis, "Metric properties of differential equations," Amer. J. Math., vol. 71, pp. 294-312, 1949.
3. B. Demidovich, "Dissipativity of a nonlinear system of differential equations," Ser. Mat. Mekh., pp. 19-27, 1961.
4. P. Hartmann, Ordinary Differential Equations. New York: Wiley, 1964.
5. W. Lohmiller and J.-J. Slotine, "Contraction analysis of nonlinear distributed systems," Int. J. Control, vol. 78, pp. 678-688, 2005.
6. J. Jouffroy, "Some ancestors of contraction analysis," in Proc. IEEE Conf. Decision Control, 2005, pp. 5450-5455.
7. E. Sontag and Y. Wang, "Output-to-state stability and detectability of nonlinear systems," Syst. Control Lett., vol. 29, pp. 279-290, 1997.
8. W. Lohmiller and J.-J. Slotine, "On contraction analysis for nonlinear systems," Automatica, vol. 34, pp. 671-682, 1998.
9. V. Fromion, G. Scorletti, and G. Ferreres, "Nonlinear performance of a PI controlled missile: An explanation," Int. J. Robust Nonlin. Control, vol. 9, pp. 485-518, 1999.
10. D. Angeli, "A Lyapunov approach to incremental stability properties," IEEE Trans. Automat. Control, vol. 47, no. 3, pp. 410-421, Mar. 2002.
11. H. Kushner, Stochastic Stability and Control. New York: Academic, 1967.
12. R. Has'minskii, Stochastic Stability of Differential Equations. Rockville, MD: Sijthoff and Nordhoff, 1980.
13. P. Florchinger, "Lyapunov-like techniques for stochastic stability," SIAM J. Control Optim., vol. 33, pp. 1151-1169, 1995.
14. X. Mao, Stability of Stochastic Differential Equations With Respect to Semimartingales. White Plains, NY: Longman, 1991.
15. P. Florchinger, "Feedback stabilization of affine in the control stochastic differential systems by the control Lyapunov function method," SIAM J. Control Optim., vol. 35, pp. 500-511, 1997.
16. J. Tsinias, "The concept of "exponential ISS" for stochastic systems and applications to feedback stabilization," Syst. Control Lett., vol. 36, pp. 221-229, 1999.
17. H. Deng, M. Krstic, and R. Williams, "Stabilization of stochastic non-linear systems driven by noise of unknown covariance," IEEE Trans. Automat. Control, vol. 46, no. 8, pp. 1237-1253, Aug. 2001.
18. T. Caraballo, P. Kloeden, and B. Schmalfuss, "Exponentially stable stationary solutions for stochastic evolution equations and their pertubation," Appl. Math. Optim., vol. 50, pp. 183-207, 2004.
19. N. Aghannan and P. Rouchon, "An intrinsic observer for a class of Lagrangian systems," IEEE Trans. Automat. Control, vol. 48, no. 6, pp. 936-945, Jun. 2003.
20. J. Jouffroy and T. Fossen, "On the combination of nonlinear contracting observers and UGES controllers for output feedback," in Proc. IEEE Conf. Decision Control, 2004, pp. 4933-4939.
21. W. Wang and J.-J. E. Slotine, "On partial contraction analysis for coupled nonlinear oscillators," Biol Cybern, vol. 92, no. 1, pp. 38-53, Jan. 2005.
22. Q.-C. Pham and J.-J. Slotine, "Stable concurrent synchronization in dynamic system networks," Neural Networks, vol. 20, no. 1, pp. 62-77, Jan. 2007.
23. B. Girard, N. Tabareau, Q.-C. Pham, A. Berthoz, and J.-J. Slotine, "Where neuroscience and dynamic system theory meet autonomous robotics: A contracting basal ganglia model for action selection," Neural Networks, vol. 21, pp. 628-641, 2008.
24. R. Horn and C. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge University Press, 1985.
25. L. Arnold, Stochastic Differential Equations: Theory and Applications New York: Wiley, 1974.
26. A. Pavlov, N. van de Wouw, and H. Nijmeijer, Uniform Output Regulation of Nonlinear Systems. Boston, MA: Birkhauser, 2006.
27. Q.-C. Pham, 2007, A Variation of Gronwall's Lemma [Online]. Available: http://arxiv.org/abs/0704.0922
28. A. Pavlov, A. Pogromsky, N. van deWouw, and H. Nijmeijer, "Convergent dynamics, a tribute to Boris Pavlovich Demidovich," Syst. Control Lett., vol. 52, pp. 257-261, Jul. 2004.
29. V. Popov, Hyperstability of Control Systems. New York: Springer-Verlag, 1973.
30. N. Tabareau and J.-J. Slotine, Notes on Contraction Theory MIT, Cambridge, MA, NSL Rep., 2005.
31. E. Dickmanns, Dynamic Vision for Intelligent Vehicles. Cambridge, MA: MIT, 1998.
32. N. Tabareau, J.-J. Slotine, and Q.-C. Pham, How Synchronization Protects From Noise 2008 [Online]. Available: http://arxiv.org/abs/0801.0011