Stability of nonlinear feedback systems: A new small-gain theorem

SIAM Journal on Control and Optimization

Volumn 46, Issue 6, 2007, Pages 1995-2012

  Chen, Anna L ^a   Chen, Gui Qiang ^b   Freeman, Randy A ^a  

^a Northwestern University   (United States)

^b NORTHWESTERN UNIVERSITY   (United States)

Author keywords

Conditional gain; Dissipativity; Feedback connection; Nonlinear systems; Performance analysis; Small gain theorem

Indexed keywords

EID: 55449085138     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012904440812     Document Type: Article

References (28)

1. B. D. O. ANDERSON, The small-gain theorem, the passivity theorem, and their equivalence, J. Franklin Inst., 298 (1972), pp. 105-115.
2. M. J. CHEN AND C. A. DESOER, Necessary and sufficient condition for robust stability of linear distributed feedback systems, Internat. J. Control, 35 (1982), pp. 255-267.
3. M. A. DAHLEH AND Y. OHTA, A necessary and sufficient condition for robust BIBO stability, Systems Control Lett., 11 (1988), pp. 271-275.
4. C. A. DESOER AND M. VIDYASAGAR, Feedback Systems: Input-Output Properties, Academic Press, New York, 1975.
5. J. C. DOYLE AND G. STEIN, Multivariable feedback design: Concepts for a classical/modern synthesis, IEEE Trans. Automat. Control, 26 (1981), pp. 4-16.
6. R. A. FREEMAN, On the necessity of the small-gain theorem in the performance analysis of nonlinear systems, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, 2001, pp. 51-56.
7. J. M. GONCALVES AND M. A. DAHLEH, Necessary condition for robust stability of a class of nonlinear systems, Automatica J. IFAC, 34 (1998), pp. 705-714.
8. D. J. HILL, Dissipative nonlinear systems: Basic properties and stability analysis, in Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, AZ, 1992, pp. 3259-3264.
9. D. J. HILL AND P. J. MOYLAN, Dissipative dynamical systems: Basic input-output and state properties, J. Franklin Inst., 309 (1980), pp. 327-357.
10. D. J. HILL AND P. J. MOYLAN, General instability results for interconnected systems, SIAM J. Control Optim., 21 (1983), pp. 256-279.
11. ∞- bounded robustness for nonlinear systems: Analysis and synthesis, IEEE Trans. Automat. Control, 48 (2003), pp. 1875-1891.
12. Z.-P. JIANG, I. MAREELS, AND Y. WANG, A Lyapunov formulation of nonlinear small gain theorem for interconnected systems, in Proceedings of the IFAC Nonlinear Control Systems Design Symposium, Tahoe City, CA, 1995, pp. 666-671.
13. Z.-P. JIANG, A. R. TEEL, AND L. PRALY, Small-gain theorem for ISS systems and applications, Math. Control Signals Systems, 7 (1994), pp. 95-120.
14. H. K. KHALIL, Nonlinear Systems, 3rd ed., Prentice-Hall, Upper Saddle River, NJ, 2002.
15. I. M. Y. MAREELS AND D. J. HILL, Monotone stability of nonlinear feedback systems, J. Math. Systems Estim. Control, 2 (1992), pp. 275-291.
16. A. MEGRETSKI AND A. RANTZER, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control, 42 (1997), pp. 819-830.
17. P. J. MOYLAN AND D. J. HILL, Tests for stability and instability of interconnected system, IEEE Trans. Automat. Control, 24 (1979), pp. 574-575.
18. J. W. POLDERMAN AND J. C. WILLEMS, Introduction to Mathematical Systems Theory: A Behavioral Approach, Springer-Verlag, New York, 1998.
19. J. S. SHAMMA, The necessary of the small-gain theorem for time-varying and nonlinear systems, IEEE Trans. Automat. Control, 36 (1991), pp. 1138-1147.
20. J. S. SHAMMA AND M. A. DAHLEH, Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization, IEEE Trans. Automat. Control, 36 (1991), pp. 838-847.
21. J. S. SHAMMA AND R. ZHAO, Fading-memory feedback systems and robust stability, Automatica J. IFAC, 29 (1993), pp. 191-200.
22. H. L. TENTELMAN AND J. C. WILLEMS, Every storage function is a stable function, Systems Control Lett., 32 (1997), pp. 249-259.
23. M. VIDYASAGAR, Input-Output Analysis of Large-Scale Interconnected Systems, Lecture Notes in Control and Inform. Sci. 29, Springer-Verlag, Berlin, 1981.
24. J. C. WILLEMS, Stability, instability, invertibility, and causality, SIAM J. Control, 7 (1969), pp. 645-671.
25. J. C. WILLEMS, Dissipative dynamical systems I: General theory, Arch. Rational Mech. Anal., 45 (1972), pp. 321-351.
26. J. C. WILLEMS, Mechanisms for the stability and instability in feedback systems, Proc. IEEE, 64 (1976), pp. 24-35.
27. J. C. WILLEMS, Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Automat. Control, 36 (1991), pp. 259-294.
28. G. ZAMES, On the input-output stability of time-varying nonlinear feedback systems I: Conditions using concepts of loop gain, conicity, and positivity, IEEE Trans. Automat. Control, 11 (1966), pp. 228-238.