Local ISS of large-scale interconnections and estimates for stability regions

Systems and Control Letters

Volumn 59, Issue 3-4, 2010, Pages 241-247

  Dashkovskiy, Sergey N ^a   Rüffer, Björn S ^b  

^a UNIVERSITY OF BREMEN   (Germany)

^b UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

Interconnected systems; Large scale system; Local input to state stability; Lyapunov function; Small gain condition

Indexed keywords

COMPOSITE SYSTEMS; INPUT TO STATE STABLE; INPUT-TO-STATE STABILITY; INTEGRAL INPUT-TO-STATE STABLE (IISS) SYSTEM; INTERCONNECTED SYSTEMS; LARGE-SCALE INTERCONNECTIONS; LOCAL INPUT-TO-STATE STABILITY; LYAPUNOV; SMALL-GAIN CONDITION; STABILITY REGIONS;

DIFFERENTIAL EQUATIONS; INTERCONNECTION NETWORKS; LARGE SCALE SYSTEMS; LYAPUNOV FUNCTIONS;

SYSTEM STABILITY;

EID: 77950023098     PISSN: 01676911     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sysconle.2010.02.001     Document Type: Article

References (24)

1. Sontag E.D. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control 34 4 (1989) 435-443
2. Jiang Z.-P., Teel A.R., and Praly L. Small-gain theorem for ISS systems and applications. Math. Control Signals Systems 7 2 (1994) 95-120
3. Jiang Z.-P., Mareels I.M.Y., and Wang Y. A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica J. IFAC 32 8 (1996) 1211-1215
4. S. Dashkovskiy, B. Rüffer, F. Wirth, A small-gain type stability criterion for large scale networks of ISS systems, in: Proc. 44th IEEE Conf. Decis. Control and Europ. Contr. Conf., Seville, Spain, Dec. 2005, pp. 5633-5638.
5. Dashkovskiy S.N., Rüffer B.S., and Wirth F.R. An ISS small-gain theorem for general networks. Math. Control Signals Systems 19 2 (2007) 93-122
6. S.N. Dashkovskiy, B.S. Rüffer, F.R. Wirth, Small gain theorems for large scale systems and construction of ISS Lyapunov functions, SIAM J. Control Optim. (September) (2009) (provisionally accepted). arXiv:0901.1842v2.
7. Liberzon D., and Nešić D. Stability analysis of hybrid systems via small-gain theorems. Hybrid Systems: Computation and Control. Lecture Notes in Comput. Sci. vol. 3927 (2006), Springer, Berlin 421-435
8. D. Nesic, A.R. Teel, A Lyapunov-based small-gain theorem for hybrid ISS systems, in: Proc. 47th IEEE Conf. Decis. Control, Cancun, Mexico, Dec. 9-11, 2008, pp. 3380-3385.
9. Pepe P., and Jiang Z.-P. A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems. Systems Control Lett. 55 12 (2006) 1006-1014
10. Karafyllis I., and Jiang Z.-P. A small-gain theorem for a wide class of feedback systems with control applications. SIAM J. Control Optim. 46 4 (2007) 1483-1517
11. Sontag E.D. Comments on integral variants of ISS. Systems Control Lett. 34 1-2 (1998) 93-100
12. Ito H. State-dependent scaling problems and stability of interconnected iISS and ISS systems. IEEE Trans. Automat. Control 51 10 (2006) 1626-1643
13. H. Ito, Zhong-Ping Jiang, Small-gain conditions and Lyapunov functions applicable equally to iISS and ISS systems without uniformity assumption, in: Proc. of the American Control Conference, ACC'08, Seattle, Washington, Jun. 11-13, 2008, pp. 2297-2303.
14. Grüne L. Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization. Lecture Notes in Mathematics vol. 1783 (2002), Springer-Verlag, Berlin
15. Sepulchre R., Janković M., and Kokotović P.V. Constructive Nonlinear Control. Communications and Control Engineering Series (1997), Springer-Verlag, Berlin
16. Sontag E.D., and Wang Y. New characterizations of input-to-state stability. IEEE Trans. Automat. Control 41 9 (1996) 1283-1294
17. Angeli D., Sontag E.D., and Wang Y. A characterization of integral input-to-state stability. IEEE Trans. Automat. Control 45 6 (2000) 1082-1097
18. S. Dashkovskiy, B. Rüffer, F. Wirth, Numerical verification of local input-to-state stability for large networks, in: Proc. 46th IEEE Conf. Decis. Control, New Orleans, USA, Dec. 12-14, 2007, pp. 4471-4476.
19. B.S. Rüffer, Monotone dynamical systems, graphs, and stability of large-scale interconnected systems, Ph.D. Thesis, Universität Bremen, Germany, Oct. 2007. Available online at: http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000109058.
20. S.N. Dashkovskiy, B.S. Rüffer, F.R. Wirth, Applications of the general Lyapunov ISS small-gain theorem for networks, in: Proc. 47th IEEE Conf. Decis. Control, Cancun, Mexico, Dec. 9-11, 2008, pp. 25-30.
21. B.S. Rüffer, Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space, Positivity (2009), in press (doi:10.1007/s11117-009-0016-5).
22. Massera J.L. Contributions to stability theory. Ann. of Math. (2) 64 (1956) 182-206
23. Chaohong Cai, A.R. Teel, Results on input-to-state stability for hybrid systems, in: Proc. 44th IEEE Conf. Decis. Control and Europ. Contr. Conf., Seville, Spain, Dec. 2005, pp. 5403-5408.
24. Karow M., Hinrichsen D., and Pritchard A.J. Interconnected systems with uncertain couplings: Explicit formulae for μ-values, spectral value sets, and stability radii. SIAM J. Control Optim. 45 3 (2006) 856-884 (electronic)