Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions

IEEE Transactions on Automatic Control

Volumn 58, Issue 5, 2013, Pages 1192-1207

  Ito, Hiroshi ^a   Jiang, Zhong Ping ^b   Dashkovskiy, Sergey N ^c   R̈uffer, Björn S ^d  

^a KYUSHU INSTITUTE OF TECHNOLOGY   (Japan)

^b POLYTECHNIC UNIVERSITY   (United States)

^c UNIVERSITY OF APPLIED SCIENCES   (Germany)

^d UNIVERSITY OF PADERBORN   (Germany)

Author keywords

Integral input to state stability (iISS); Large scale systems; Lyapunov functions; Nonlinear systems; Robust stability criteria

Indexed keywords

LARGE SCALE SYSTEMS; NONLINEAR SYSTEMS; ROBUSTNESS (CONTROL SYSTEMS); STABILITY CRITERIA;

GENERAL STRUCTURES; INPUT TO STATE STABLE; INTEGRAL INPUT-TO-STATE STABILITY (IISS); INTEGRAL INPUTTOSTATE STABLE (IISS); ROBUST STABILITY; SMALL GAIN THEOREM; SMALL-GAIN CONDITION; STABILITY PROBLEM;

LYAPUNOV FUNCTIONS;

EID: 84883542657     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2012.2231552     Document Type: Article

References (40)

1. E. D. Sontag, "Smooth stabilization implies coprime factorization," IEEE Trans. Autom. Control, vol. 34, no. 4, pp. 435-443, Apr. 1989.
2. Z. P. Jiang, A. R. Teel, and L. Praly, "Small-gain theorem for ISS systems and applications," Math. Control Signals Syst., vol. 7, pp. 95-120, 1994.
3. A. R. Teel, "A nonlinear small gain theorem for the analysis of control systems with saturation," IEEE Trans. Autom. Control, vol. 41, no. 9, pp. 1256-1270, Sep. 1996.
4. E. D. Sontag, "Comments on integral variants of ISS," Syst. Control Lett., vol. 34, pp. 93-100, 1998.
5. A. Goldbeter, "A model for circadian oscillations in the Drosophila period protein (PER)," Royal Soc. Lond. B., vol. 261, pp. 319-324, 1995.
6. F. Mazenc and M. Malisoff, "Strict Lyapunov function constructions under LaSalle conditions with an application to Lotka-Volterra systems," IEEE Trans. Autom. Control, vol. 55, pp. 84-854, 2010.
7. H. Ito, "Asymmetric iISS and ISS properties yielding stability of interconnected systems and an application to a circadian model," in Proc. American Control Conf., 2007, pp. 3041-3046.
8. I. Karafyllis and Z. P. Jiang, "A new small-gain theorem with an application to the stabilization of the chemostat," Int. J. Robust Nonlinear Control, vol. 22, no. 14, pp. 1602-1630, 2012.
9. H. Ito, H. Suzuki, and S. Taketomi, "Delay robustness of CDMApower control in the presence of disturbances using iISS small-gain technique," in Proc. IEEE Conf. Control Appl., 2010, pp. 1109-1114.
10. C. Agnew, "Dynamic modeling and control of congestion-prone systems," Oper. Research, vol. 24, pp. 400-419, 1976.
11. D. Tipper and M. K. Sundareshan, "Numerical methods for modeling computer networks under non-stationary conditions," IEEE JSAC, vol. 8, pp. 1682-1695, 1990.
12. J. Le Ny and H. Balakrishnan, "Feedback control of the national airspace system," J. Guid., Control Dyn., vol. 34, pp. 832-846, 2011.
13. M. Arcak, D. Angeli, and E. Sontag, "A unifying integral ISS framework for stability of nonlinear cascades," SIAM J. Control Optim., vol. 40, no. 6, pp. 1888-1904, 2002.
14. H. Ito, "State-dependent scaling problems and stability of interconnected iISS and ISS systems," IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1626-1643, Oct. 2006.
15. D. Angeli and A. Astolfi, "A tight small gain theorem for not necessarily ISS systems," Syst. Control Lett., vol. 56, pp. 87-91, 2007.
16. A. Chaillet and D. Angeli, "Integral input to state stable systems in cascade," Syst. Control Lett., vol. 57, pp. 519-527, 2008.
17. S. Dashkovskiy, M. Kosmykov, and F. R. Wirth, "A small-gain condition for interconnections of ISS systems with mixed ISS characterizations," IEEE Trans. Autom. Control, vol. 56, pp. 1247-1258, 2011.
18. S. Dashkovskiy, B. Rüffer, and F. Wirth, "An ISS small-gain theorem for general networks," Math. Control Signals Syst., vol. 19, no. 2, pp. 93-122, 2007.
19. Z. P. Jiang and Y. Wang, "A generalization of the nonlinear small-gain theorem for large-scale complex systems," in Proc. 2008 World Congress Intell. Control Autom., 2008, pp. 1188-1193.
20. T. Liu, D. J. Hill, and Z. P. Jiang, "Lyapunov formulation of ISS small-gain in continuous-time dynamical networks," Automatica, vol. 47, pp. 2088-2093, 2011.
21. S. Dashkovskiy, B. Rüffer, and F.Wirth, "Small gain theorems for large scale systems and construction of ISS Lyapunov functions," SIAM J. Control Optim., vol. 48, no. 6, pp. 4089-4118, 2010.
22. I. Karafyllis and Z. P. Jiang, "A vector small-gain theorem for general nonlinear control systems," IMA J. Math. Control Info., vol. 28, no. 3, pp. 309-344, 2011.
23. S. Dashkovskiy, H. Ito, and F.Wirth, "On a small-gain theorem for ISS networks in dissipative Lyapunov form," Eur. J. Control, vol. 17, no. 4, pp. 357-369, 2011.
24. Z. P. Jiang, I. Mareels, and Y. Wang, "A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems," Automatica, vol. 32, pp. 1211-1215, 1996.
25. H. Ito, S. Dashkovskiy, and F. Wirth, "On a small gain theorem for networks of iISS systems," in Proc. 48th IEEE Conf. Decision Control, 2009, pp. 4210-4215.
26. D. D. Šiljak, Large-Scale Dynamic Systems: Stability and Structure. New York: North-Holland, 1978.
27. N. R. Sandell, P. Varaiya, M. Athans, and M. G. Safonov, "Survey of decentralized control methods for large scale systems," IEEE Trans. Autom. Control, vol. 23, no. 2, pp. 108-128, Feb. 1978.
28. D. D. Šiljak, Decentralized Control of Complex Systems. Boston, MA: Academic Press, 1991.
29. D. J. Hill and P. J. Moylan, "Stability results for nonlinear feedback systems," Automatica, vol. 13, pp. 377-382, 1977.
30. H. Ito and Z. P. Jiang, "Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective," IEEE Trans. Autom. Control, vol. 54, no. 10, pp. 2389-2404,Oct. 2009.
31. H. Ito, "A Lyapunov approach to cascade interconnection of integral input-to-state stable systems," IEEE Trans. Autom. Control, vol. 55, no. 3, pp. 702-708, Mar. 2010.
32. H. Ito, "Construction of Lyapunov functions for networks of iISS systems: An explicit solution for a cyclic structure," in Proc. American Control Conf., 2010, pp. 196-201.
33. B. S. Rüffer, C. M. Kellett, and S. R.Weller, "Connection between cooperative positive systems and integral input-to-state stability of largescale systems," Automatica, vol. 46, no. 6, pp. 1019-1027, 2010.
34. H. Ito, "Necessary conditions for global asymptotic stability of networks of iISS systems," Math. Control Signals Syst., vol. 24, no. 1, pp. 55-74, 2012.
35. D. Angeli, E. D. Sontag, and Y. Wang, "A characterization of integral input-to-state stability," IEEE Trans. Automat. Control, vol. 45, no. 6, pp. 1082-1097, Jun. 2000.
36. E. D. Sontag and Y. Wang, "On characterizations of input-to-state stability property," Syst. Control Lett., vol. 24, pp. 351-359, 1995.
37. E. D. Sontag and A. R. Teel, "Changing supply functions in input/state stable systems," IEEE Trans. Autom. Control, vol. 40, pp. 1476-1478, 1995.
38. R. J. Wilson and J. J. Watkins, Graphs: An Introductory Approach. New York: Wiley, 1990.
39. H. Ito, "Connecting several stability criteria for iISS networks and their application to a network computing model," in Proc. 50th IEEE Conf. Decision Control & Eur. Control Conf., 2011, pp. 6072-6078.
40. B. S. Rüffer, "Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space," Positivity, vol. 14, no. 2, pp. 257-283, 2010.