Converse Lyapunov theorems and robust asymptotic stability for hybrid systems

Proceedings of the American Control Conference

Volumn 1, Issue , 2005, Pages 12-17

  Cai, Chaohong ^a   Teel, Andrew R ^a   Goebel, Rafal ^a  

^a University of California   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL INCLUSIONS; LYAPUNOV FUNCTIONS; LYAPUNOV THEOREMS; NATURAL EXTENSION;

ASYMPTOTIC STABILITY; DIFFERENTIAL EQUATIONS; FUNCTIONS; LYAPUNOV METHODS; ROBUSTNESS (CONTROL SYSTEMS); THEOREM PROVING;

HYBRID COMPUTERS;

EID: 23944504793     PISSN: 07431619     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (29)

1. J.-P. Aubin, J. Lygeros, M. Quincampoix, S. Sastry S. and N. Seube. Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Auto. Cont., 47:2-20, 2002.
2. M.S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Auto. Cont., 43(4):475-482, 1998.
3. M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. Solution concepts for hybrid dynamical systems. In 15th IFAC World Congress on Automatic Control, pages 1217-1222, 2002.
4. F. H. Clarke, Y. S. Ledyaev, and R. J. Stern. Asymptotic stability and smooth Lyapunov functions. J. Diff. Equations, 149:69-114, 1998.
5. P. Collins. A trajectory-space approach to hybrid systems. In MTNS. Springer, Berlin, 2004.
6. R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson. Perspectives and results on the stability and stabilizability of hybrid systems. In Proc. IEEE, volume 88, pages 1069-1082, 2000.
7. R. Goebel, J. Hespanha, A.R. Teel, C. Cai, and R. Sanfelice. Hybrid systems: generalized solutions and robust stability. In IFAC Symp. Nonlinear Control Systems, Stuttgart, Germany, September 2004.
8. R. Goebel and A. R. Teel. Results on solution sets to hybrid systems with applicatino to stability theory. In Proc. 24th ACC, 2005.
9. C. M. Kellett and A.R. Teel. Smooth Lyapunov functions and robustness of stability for difference inclusions. Sys. & Cont. Lett., 52:395-405, 2004.
10. C.M. Kellett and A.R. Teel. Results on converse Lyapunov theorems for difference inclusions. In Proc. 42nd IEEE CDC, pages 3627-3632, 2003.
11. J. Kurzweil. On the inversion of Ljapunov's second theorem on stability of motion. Amer. Math. Soc. Trans., Ser. 2, 24:19-77, 1956.
12. V. Lakshmikantham, S. Leela, and A.A. Martynyuk. Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., 1989.
13. V. Lakshmikantham, S. Leela, and A. S. Vatsala. Setvalued hybrid differential equations and stability in terms of two measures. Inter. J. Hybrid Systems, 2:169-188, 2002.
14. Y.S. Ledyaev and E.D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Analysis, 37:813-840, 1999.
15. D. Liberzon. Switching in Systems and Control. Birkhauser, 2003.
16. Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Cont. & Opt., 34:124-160, 1996.
17. J. Lygeros, K.H. Johansson, S. Sastry, and M. Egerstedt. On the existence of executions of hybrid automata. In Proc. 38th IEEE CDC, pages 2249-2254, 1999.
18. J. Lygeros, K.H. Johansson, S.N. Simić, J. Zhang, and S.S. Sastry. Dynamical properties of hybrid automata. IEEE Trans. Auto. Cont., 48(1):2-17, 2003.
19. J. L. Mancilla-Aguilar and R. A. Garcia. A converse Lyapunov theorem for nonlinear switched systems. Sys. & Cont. Lett., 41(1):67-71, 2002.
20. A.N. Michel and B. Hu. Towards a stability theory of general hybrid dynamical systems. Automatica, 35:371-384, 1999.
21. A.A. Movchan. Stability of processes with respect to two measures. Prikl. Mat. Mekh., pages 988-1001, 1960.
22. C. Prieur. Perturbed hybrid systems, applications to control theory. In Nonlinear & Adaptive Cont., pages 285-294. Springer, Berlin, 2003.
23. C. Prieur and A. Astolfi. Robust stabilization of chained systems via hybrid control. IEEE Trans. Auto. Cont., 48(10): 1768-1772, 2003.
24. R. G. Sanfelice, R. Goebel. and A. R. Teel. Results on convergence in hybrid systems via detectability and an invariance principle. In Proc. 24th ACC, 2005.
25. E.D. Sontag and Y. Wang. Notions of input to output stability. Sys. & Cont. Lett., 38:235-248, 1999.
26. L. Tavernini. Differential automata and their simulators. Nonlinear Analysis, Theory, Methods and Applications, 11(6):665-683, 1987.
27. A. R. Teel and L. Praly. A smooth Lyapunov function from a class-K script L sign estimate involving two positive simidefinite functions. ESAIM: Control, Optimisation and Calculus of Variations, 5:313-367, 2000.
28. A. Van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems. Springer, 2000.
29. H. Ye, A.N. Michel, and L. Hou. Stability theory for hybrid dynamical systems. IEEE Trans. Auto. Cont., 43(4):461-474, 1998.