Numerical construction of LISS Lyapunov functions under a small gain condition

Proceedings of the IEEE Conference on Decision and Control

Volumn , Issue , 2011, Pages 6967-6972

  Geiselhart, Roman ^a   Wirth, Fabian ^a  

^a UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

homotopy algorithm; interconnected system; LISS Lyapunov function; monotone operator; small gain condition

Indexed keywords

CONTROL ENGINEERING; LARGE SCALE SYSTEMS;

FIXED POINTS; FIXED-POINT ALGORITHMS; HOMOTOPY ALGORITHMS; INPUT-TO-STATE STABILITY; LISS LYAPUNOV FUNCTIONS; MONOTONE OPERATORS; SEMI-GLOBAL; SMALL-GAIN CONDITION;

LYAPUNOV FUNCTIONS;

EID: 84860672640     PISSN: 07431546     EISSN: 25762370     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2011.6160908     Document Type: Conference Paper

References (17)

1. C. Berge. Topological Spaces. Dover Pub., Mineola, NY, 1997.
2. S. Dashkovskiy, H. Ito, and F. Wirth. On a small gain theorem for ISS networks in dissipative Lyapunov form. European J. Control, 2011. to appear.
3. S. Dashkovskiy and B. Rüffer. Local ISS of large-scale interconnections and estimates for stability regions. Systems Control Lett., 59(3-4):241-247, 2010.
4. S. Dashkovskiy, B. Rüffer, and F. Wirth. An ISS small gain theorem for general networks. Math. Control, Signals, and Syst., 19:93-122, 2007.
5. S. Dashkovskiy, B. Rüffer, and F. Wirth. Small gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM J. Control Optim., 48:4089-4118, 2010.
6. B. Eaves. Homotopies for computation of fixed points. Math. Programming, 3:1-22, 1972.
7. R. Geiselhart and F. Wirth. Numerical construction of LISS Lyapunov functions under a small gain condition. submitted, 2011. preprint on arXiv:1105.1922v1.
8. H. Ito and Z. Jiang. Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective. IEEE Trans. Automatic Control, 54(10):2389-2404, 2009.
9. 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.
10. Z.-P. Jiang, A. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Math. Control Signals System, 7(2):95-120, 1994.
11. T. Liu, D. Hill, and Z. Jiang. Lyapunov formulation of ISS small-gain in dynamical networks. In Proc. 48th IEEE CDC/28th CCC 2009, pages 4204-4209, Shanghai, 2009. IEEE.
12. B. Rüffer. Small-gain conditions and the comparison principle. IEEE Trans. Automatic Control, 55(7):1732-1736, 2010.
13. B. S. Rüffer. Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space. Positivity, 14(2):257-283, 2010.
14. B. S. Rüffer, P. M. Dower, and H. Ito. Applicable comparison principles in large-scale system analysis. In Proc. of the 10th SICE Annual Conference on Control Systems, Kumamoto, Japan, Mar. 2010.
15. B. S. Rüffer and F. R. Wirth. Stability verification for monotone systems using homotopy algorithms. Numerical Algorithms, 2011. to appear, available at arXiv:1005.0741.
16. E. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435-443, 1989.
17. Z. Yang. Computing Equilibria and Fixed Points. Kluwer Academic Pub., Boston, Dordrecht, London, 1999.