Stabilization of switched systems via composite quadratic functions

IEEE Transactions on Automatic Control

Volumn 53, Issue 11, 2008, Pages 2571-2585

  Hu, Tingshu ^a   Ma, Liqiang ^a   Lin, Zongli ^b  

^a University of Massachusetts   (United States)

^b UNIVERSITY OF VIRGINIA   (United States)

Author keywords

Bilinear matrix inequality (BMI); Composite quadratic functions; Sliding mode; Stabilization

Indexed keywords

DIFFERENTIAL EQUATIONS; FEEDBACK CONTROL; FUNCTIONS; LINEAR CONTROL SYSTEMS; LYAPUNOV FUNCTIONS; METALLIC MATRIX COMPOSITES; PROBABILITY DENSITY FUNCTION; SIGNAL RECONSTRUCTION; STABILIZATION; SWITCHING FUNCTIONS; SWITCHING NETWORKS; SWITCHING SYSTEMS; TELECOMMUNICATION NETWORKS;

BILINEAR MATRIX INEQUALITY (BMI); COMPOSITE QUADRATIC FUNCTIONS; CONVEX HULLS; DIFFERENTIABLE FUNCTIONS; DIRECTIONAL DERIVATIVES; EXISTING CONDITIONS; NONCONVEX; NUMERICAL EXAMPLES; NUMERICAL TOOLS; QUADRATIC FUNCTIONS; SLIDING MODE; SUB SYSTEMS; SWITCHED SYSTEMS; SWITCHING LAWS; TWO TYPES;

NUMBER THEORY;

EID: 58249135267     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2008.2006933     Document Type: Article

References (39)

1. F. Blanchini and C. Savorgnan, "Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions," in Proc. 45th IEEE Conf. Decision Control, 2006, pp. 119-124.
2. S. Boyd, Lecture Notes for Convex Optimization II [Online]. Available: http://www.stanford.edu/class/ee364b/lectures/subgradients_slides.pdf
3. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA: SIAM, 1994.
4. M. S. Branicky, "Multiple Lyapunov function and other analysis tools for switched and hybrid systems," IEEE Trans. Automat. Control, vol. 43, no. 4, pp. 475-482, Apr. 1998.
5. M. S. Branicky, "Stability of switched and hybrid systems: State of the art," in Proc. 36th IEEE Conf. Decision Control, 1997, pp. 1208-1212.
6. G. Chesi, A. Garulli, A. Tesi, and A. Vicino, "Homogeneous Lyapunov functions for systems with structured uncertainties," Automatica, vol. 39, pp. 1027-1035, 2003.
7. W. P. Dayawansa and C. F. Martin, "A converse Lyapunov theorem for a class of dynamical systems which undergo switching," IEEE Trans. Automat. Control, vol. 44, no. 4, pp. 751-760, Apr. 1999.
8. R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, "Perspectives and results on the stability and stabilizability of hybrid systems," Proc. IEEE, vol. 88, no. 7, pp. 1069-1082, Jul. 2000.
9. E. Feron, "Quadratic Stabilizability of Switched Systems via State and Output Feedback," Center for Intelligent Control Systems, MIT, Cambridge, MA, Tech. Rep. CICS P-468, 1996.
10. A. F. Filippov, Differential Equations With Discontinuous Right Hand Sides. Norwell, MA: Kluwer Academic, 1988.
11. R. Goebel, T. Hu, and A. R. Teel, "Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions," in Current Trends in Nonlinear Systems and Control. Basel, Switzerland: Birkhauser, 2005.
12. R. Goebel, A. R. Teel, T. Hu, and Z. Lin, "Dissipativity for dual linear differential inclusions through conjugate storage functions," in Proc. IEEE Conf. Dec. Control, 2004, pp. 2700-2705.
13. A. Hassibi, J. How, and S. Boyd, "A path-following method for solving BMI problems in control," in Proc. Amer. Control Conf., 1999, pp. 1385-1389.
14. B. Hu, X. Xu, A. N. Michel, and P. J. Antsaklis, "Robust stabilizing control laws for a class of second-order switched systems," Syst. Control Lett., vol. 38, pp. 197-207, 1999.
15. B. Hu, G. Zhai, and A. N. Michel, "Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems," Int. J. Control, vol. 75, no. 14, pp. 1127-1135, 2002.
16. T. Hu, "Nonlinear control design for linear differential inclusions via convex hull quadratic Lyapunov functions," Automatica, vol. 43, no. 4, pp. 685-692, 2007.
17. T. Hu, R. Goebel, A. R. Teel, and Z. Lin, "Conjugate Lyapunov functions for saturated linear systems," Automatica, vol. 41, no. 11, pp. 1949-1956, 2005.
18. T. Hu and Z. Lin, "Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions," IEEE Trans. Automat. Control, vol. 50, no. 6, pp. 781-798, Jun. 2005.
19. T. Hu and Z. Lin, "Composite quadratic Lyapunov functions for constrained control systems," IEEE Trans. Automat. Control, vol. 48, no. 3, pp. 440-450, Mar. 2003.
20. T. Hu, A. R. Teel, and L. Zaccarian, "Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions," IEEE Trans. Automat. Control, vol. 51, no. 11, pp. 1770-1786, Nov. 2006.
21. Z. Jarvis-Wloszek and A. K. Packard, "An LMI method to demonstrate simultaneous stability using nonquadratic polynomial Lyapunov functions," in Proc. IEEE Conf. Decision Control, 2002, pp. 287-292.
22. M. Johansson and A. Rantzer, "Computation of piecewise quadratic Lyapunov functions for hybrid systems," IEEE Trans. Automat. Control vol. 43, no. 4, pp. 555-559, Apr. 1998.
23. D. Liberzon, Switching in Systems and Control. Basel, Switzerland: Birkhauser, 2003.
24. D. Liberzon and A. S. Morse, "Basic problems in stability and design os switched systems," IEEE Control Syst. Mag., vol. 19, no. 5, pp. 59-70, 1999.
25. H. Lin and P. Antsaklis, "Stability and stabilizability of switched linear systems: A short survey of recent results," in Proc. IEEE Int. Symp. Intell. Control, Limassol, Cyprus, 2005, pp. 24-29.
26. A. N. Michel, "Recent trends in the stability analysis of hybrid dynamical systems," IEEE Trans. Circuits Syst. I, vol. 46, no. 1, pp. 120-134, Jan. 1999.
27. A. P. Molchanov, "Criteria of asymptotic stability of differential and difference inclusions encountered in control theory," Syst. Control Lett., vol. 13, pp. 59-64, 1989.
28. P. Peleties and R. A. DeCarlo, "Asymptotic stability of m-switched systems using Lyapunov functions," in Proc. IEEE Conf. Decision Control, 1992, pp. 3438-3439.
29. S. Pettersson, "Controller design of switched linear systems," in Proc. Amer. Control Conf., 2004, pp. 3869-3874.
30. S. Pettersson, "Synthesis of switched linear systems handling sliding motions," in Proc. IEEE Int. Symp. Intell. Control, Med. Conf. Control Automat., 2005, pp. 18-23.
31. R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970.
32. V. I. Utkin, "Variable structure systems with sliding mode," IEEE Trans. Automat. Control, vol. AC-22, no. 2, pp. 212-222, Feb. 1977.
33. M. A. Wicks, P. Peleties, and R. A. Decarlo, "Construction of piecewise Lyapunov functions for stabilizing switched systems," in Proc. 33rd IEEE Conf. Decision Control, Lake Buena Vista, FL, 1994, pp. 3492-3497.
34. M. A.Wicks, P. Peletics, and R. A. Decarlo, "Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems," Eur. J. Control, vol. 4, pp. 140-147, 1998.
35. M.Wicks and R. Decarlo, "Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems," in Proc. Amer. Control Conf., 1997, pp. 1709-1713.
36. L. Xie, S. Shishkin, and M. Fu, "Piecewise Lyapunov functions for robust stability of linear time-varying systems," Syst. Control Lett., vol. 31, pp. 165-171, 1997.
37. X. Xu and P. J. Antsaklis, "Stabilization of second-order LTI switched systems," Int. J. Control, vol. 73, pp. 1216-1279, 2000, pp. 1339-1344, 1999.
38. H. Ye, A. N. Michel, and L. Hou, "Stability for hybrid dynamical systems," IEEE Trans. Automat. Control, vol. 43, no. 4, pp. 461-474, Apr. 1998.
39. G. Zhai, H. Z. Lin, and P. J. Antsaklis, "Quadratic stabilizability of switched linear systems with polytopic uncertainties," Int. J. Control, vol. 76, no. 7, pp. 747-753, 2003.