Applications of the general Lyapunov ISS small-gain theorem for networks

Proceedings of the IEEE Conference on Decision and Control

Volumn , Issue , 2008, Pages 25-30

  Dashkovskiy, Sergey N ^a   Rüffer, Björn S ^b   Wirth, Fabian R ^c  

^a UNIVERSITY OF BREMEN   (Germany)

^b UNIVERSITY OF NEWCASTLE   (Australia)

^c UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Constructive method; Example applications; General small gain theorem; Input to state stability (ISS); Lyapunov functions

Indexed keywords

LINEAR SYSTEMS; NEURONS;

CONSTRUCTIVE METHODS; EXPLICIT CONSTRUCTIONS; FORMATION CONTROL; INPUT-TO-STATE STABILITY; ISS-LYAPUNOV FUNCTIONS; LINEAR RESISTORS; NEURONAL NETWORKS; SMALL GAIN THEOREM;

LYAPUNOV FUNCTIONS;

EID: 62949151536     PISSN: 07431546     EISSN: 25762370     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2008.4739299     Document Type: Conference Paper

References (22)

1. F. H. Clarke. Nonsmooth analysis in control theory: a survey. European J. Control, 7:63-78, 2001.
2. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski. Nonsmooth analysis and control theory. Springer-Verlag, New York, 1998.
3. S. Dashkovskiy, B. Rüffer, and F. Wirth. A small-gain type stability criterion for large scale networks of ISS systems. In Proc. of 44th IEEE Conference on Decision and Control and European Control Conference CDC/ECC 2005, pages 5633-5638, Seville, Spain, December 2005.
4. S. Dashkovskiy, B. Rüffer, and F. Wirth. An ISS Lyapunov function for networks of ISS systems. In Proc. 17th Int. Symp. Math. Theory of Networks and Systems, MTNS2006, pages 77-82, Kyoto, Japan, July 24-28 2006.
5. S. Dashkovskiy, B. Rüffer, and F. Wirth. An ISS small-gain theorem for general networks. Math. Control Signals Systems, 19:93-122, 2007.
6. S. Dashkovskiy, B. Rüffer, and F. Wirth. Numerical verification of local input-to-state stability for large netwoks. In Proc. of 46th IEEE Conference on Decision and Control, CDC 2007, pages 4471-4476, 12-14 December, 2007, New Orleans, USA, December 2007.
7. S. Dashkovskiy, B. S. Rüffer, and F. R. Wirth. A Lyapunov ISS small gain theorem for strongly connected networks. In Proc. 7th IFAC Symposium on Nonlinear Control Systems, NOLCOS2007, pages 283-288, Pretoria, South Africa, August 2007.
8. S. Dashkovskiy, B. S. Rüffer, and F. R. Wirth. Stability of autonomous logistic processes via small gain criteria. In Proc. International Conference on Dynamics in Logistics, pages 359-366, Bremen, August 2007.
9. S. N. Dashkovskiy, B. S. Rüffer, and F. R. Wirth. Construction of an iss-lyapunov function for large-scale interconnections of iss systems. In preparation, 2008.
10. D. Hinrichsen and A. J. Pritchard. Composite systems with uncertain couplings of fixed structure: Scaled riccati equations and the problem of quadratic stability. SIAM J. Control Optim., 2008. to appear.
11. Z.-P. Jiang, I. M. Y. Mareels, and Y. Wang. A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica J. IFAC, 32(8):1211-1215, 1996.
12. Z.-P. Jiang, A. R. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Math. Control Signals Systems, 7(2):95-120, 1994.
13. M. Karow, D. Hinrichsen, and A. J. Pritchard. Spectral value sets, stability radii and μ-functions for composite systems. SIAM J. Control Optim., 45(3):846-884, 2006.
14. A. N. Michel, R. K. Miller, and W. Tang. Lyapunov stability of interconnected systems: decomposition into strongly connected subsystems. IEEE Trans. Circuits and Systems, 25(9):799-809, 1978.
15. B. S. Rüffer. Monotone Systems, Graphs, and Stability of Large-Scale Interconnected Systems. Dissertation, Fachbereich 3, Mathematik und Informatik, Universität Bremen, Germany, August 2007. Available online: http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000109058.
16. B. S. Rüffer. Monotone inequalities, dynamical systems, and paths in the positive orthant of euclidean n-space. Positivity, 2008. submitted.
17. I. W. Sandberg. Some theorems on the dynamic response of nonlinear transistor networks. Bell System Tech. J., 48:35-54, 1969.
18. R. Sepulchre, M. Janković, and P. V. Kokotović. Constructive nonlinear control. Communications and Control Engineering Series. Springer-Verlag, Berlin, 1997.
19. Eduardo D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989.
20. Eduardo D. Sontag and Yuan Wang. New characterizations of inputto-state stability. IEEE Trans. Automat. Control, 41(9):1283-1294, 1996.
21. H. G. Tanner, G. J. Pappas, and V. Kumar. Leader-to-formation stability. IEEE Transactions on Robotics and Automation, 20(3):443-455, 2004.
22. Lin Wang and Xingfu Zou. Exponential stability of Cohen-Grossberg neural networks. Neural networks, 15:415-422, 2002.