Generating functions of switched linear systems: Analysis, computation, and stability applications

IEEE Transactions on Automatic Control

Volumn 56, Issue 5, 2011, Pages 1059-1074

  Hu, Jianghai ^a   Shen, Jinglai ^b   Zhang, Wei ^c  

^a PURDUE UNIVERSITY   (United States)

^b UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Optimal control; stability; switched systems

Indexed keywords

ARBITRARY SWITCHING; DISCRETE-TIME; EXPONENTIAL GROWTH RATES; EXPONENTIAL STABILITY; GENERATING FUNCTIONS; INITIAL STATE; OPTIMAL CONTROLS; OPTIMAL SWITCHING; RANDOM SWITCHING; SUFFICIENT CONDITIONS; SWITCHED LINEAR SYSTEM; SWITCHED SYSTEMS; SWITCHING RULE; SYSTEM TRAJECTORY; UNIFIED FRAMEWORK;

CONVERGENCE OF NUMERICAL METHODS; DISCRETE TIME CONTROL SYSTEMS; EXPONENTIAL FUNCTIONS; LINEAR SYSTEMS; OPTIMIZATION; STABILITY; SWITCHING; SWITCHING FUNCTIONS; SWITCHING SYSTEMS;

CONTROL SYSTEM STABILITY;

EID: 79955926904     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2010.2067590     Document Type: Article

References (47)

1. D. Liberzon, Switching in Systems and Control. Cambridge, : Birkhäuser, 2003.
2. H. Lin and P. Antsaklis, "Stability and stabilizability of switched linear systems: A survey of recent results," IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308-322, Feb. 2009.
3. R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, "Stability criteria for switched and hybrid systems," SIAM Rev., vol. 49, no. 4, pp. 545-592, 2007. (Pubitemid 350263542)
4. J. Hespanha and A. S. Morse, "Stability of switched systems with average dwell time," in Proc. 38th IEEE Int. Conf. Decision Control, Phoenix, AZ, 1999, pp. 2655-2660. (Pubitemid 30575356)
5. M. Johansson and A. Rantzer, "Computation of piecewise quadratic Lyapunov functions for hybrid systems," IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 555-559, Apr. 1998. (Pubitemid 128737378)
6. J. Shen and J. Hu, "Stability of discrete-time conewise linear inclusions and switched linear systems," in Proc. Amer. Control Conf., Baltimore, MD, 2010, pp. 4034-4039.
7. M. Branicky, "Multiple Lyapunov functions and other analysis tools for switched and hybrid systems," IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475-482, Apr. 1998. (Pubitemid 128737371)
8. T. Hu, L. Ma, and Z. Lin, "Stabilization of switched systems via composite quadratic functions," IEEE Trans. Autom. Control, vol. 53, no. 11, pp. 2571-2585, Dec. 2008.
9. A. Agrachev and D. Liberzon, "Lie-algebraic conditions for exponential stability of switched systems," in Proc. 38th IEEE Int. Conf. Decision Control, Phoenix, AZ, 1999, pp. 2679-2684. (Pubitemid 30575360)
10. D. Liberzon, J. Hespanha, and A. S. Morse, "Stability of switched linear systems: A Lie-algebraic condition," Syst. Control Lett., vol. 37, no. 3, pp. 117-122, 1999. (Pubitemid 129608440)
11. Y. Sharon and M. Margaliot, "Third-order nilpotent, nice reachability and asymptotic stability," J. Differ. Equ., vol. 233, pp. 136-150, 2007. (Pubitemid 44824770)
12. S. Pettersson and B. Lennartson, "LMI for stability and robustness of hybrid systems," in Proc. Amer. Control Conf., Albuquerque, NM, 1997, pp. 1714-1718.
13. D. Mignone, G. Ferrari-Trecate, and M. Morari, "Stability and stabilization of piecewise affine and hybrid systems: An LMI approach," in Proc. 39th IEEE Int. Conf. Decision Control, Sydney, Australia, 2000, pp. 504-509. (Pubitemid 32528773)
14. A. Rantzer and M. Johansson, "Piecewise linear quadratic optimal control," IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 629-637, Apr. 2000.
15. N. Barabanov, "Lyapunov exponent and joint spectral radius: Some known and new results," in Proc. 44th IEEE Int. Conf. Decision Control, Seville, Spain, 2005, pp. 2332-2337. (Pubitemid 46297319)
16. L. Gurvits, "Stability of discrete linear inclusions," Linear Algebra Appl., vol. 231, pp. 47-85, 1995.
17. R. Jungers, The Joint Spectral Radius, Theory and Applications. Berlin, Germany: Springer-Verlag, 2009.
18. J. Theys, "Joint spectral radius: Theory and approximations," Ph.D. dissertation, Dept. Ing. Math., Universite Catholique de Louvain, Louvain- la-Neuve, Belgium, 2005.
19. F.Wirth, "The generalized spectral radius and extremal norms," Linear Algebra Appl., vol. 342, no. 1-3, pp. 17-40, 2002.
20. V. Blondel and J. Tsitsiklis, "The boundedness of all products of a pair of matrices is undecidable," Syst. Control Lett., vol. 41, pp. 135-140, 2000.
21. R. Jungers, V. Protasov, and V. Blondel, "Overlap-free words and spectra of matrices," Theor. Comput. Sci., pp. 3670-3684, 2009.
22. L. Arnold, Random Dynamical Systems. Berlin, Germany: Springer- Verlag, 1998.
23. N. Barabanov, "A method for calculating the Lyapunov exponent of a differential inclusion," Autom. Remote Control, vol. 50, pp. 475-479, 1989.
24. J. Tsitsiklis and V. Blondel, "The Lyapunov exponent and joint spectral radius of pairs of matrices are hard-when not impossible-to compute and to approximate," Math. Control, Signals, Syst., vol. 10, pp. 31-41, 1997.
25. V. Blondel, J. Theys, and Y. Nesterov, "Approximations of the rate of growth of switched linear systems," in Hybrid Systems: Comput. and Control. Philadelphia, PA: Springer-Verlag, 2004, pp. 173-186.
26. V. Blondel, Y. Nesterov, and J. Theys, "On the accuracy of the ellipsoid norm approximation of the joint spectral radius," Linear Algebra Appl., vol. 394, pp. 91-107, 2005.
27. V. Kozyakin, "Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets," Discret. Cont. Dynam. Syst. Series B, vol. 14, pp. 143-158, 2010.
28. P. Parrilo and A. Jadbabaie, "Approximation of the joint spectral radius using sum of squares," Linear Algebra Appl., vol. 428, no. 10, pp. 2385-2402, 2008.
29. V. Protasov, R. Jungers, and V. Blondel, "Joint spectral characteristics of matrices: A conic programming approach," SIAM J. Matrix Anal. Appl., vol. 31, no. 4, pp. 2146-2162, 2010.
30. L. Fainshil, M. Margaliot, and P. Chigansky, "On the stability of positive linear switched systems under arbitrary switching laws," IEEE Trans. Autom. Control, vol. 54, no. 4, pp. 897-899, Apr. 2009.
31. J. Shen and J. Hu, "Stability of switched linear systems on cones: A generating function approach," in Proc. 49th IEEE Int. Conf. Decision Control, Atlanta, GA, 2010, pp. 420-425.
32. M. Margaliot, "Stability analysis of switched systems using variational principles: An introduction," Automatica, vol. 42, pp. 2059-2077, 2006. (Pubitemid 44647189)
33. E. Pyatnitsky, "Absolute stability of nonstationary nonlinear systems," Autom. Remote Control, vol. 31, no. 1, pp. 5-15, 1970.
34. A. Molchanov and Y. S. Pyatnitsky, "Criteria of asymptotic stability of differential and difference inclusions encountered in control theory," Syst. Control Lett., vol. 13, no. 1, pp. 59-64, 1989.
35. D. Angeli, "A note on stability of arbitrarily switched homogeneous systems," 1999.
36. X. Feng, "Lyapunov exponents and stability of linear stochastic systems," Ph.D. dissertation, Dept. Syst. Control Eng., Case Western Reserve Univ., Cleveland, OH, 1990.
37. S. Krantz and H. Parks, A Primer of Real Analytic Functions, 2nd ed. Boston, MA: Birkhäuser-Verlag, 2002.
38. M. Margaliot and M. Branicky, "Nice reachability for planar bilinear control systems with applications to planar linear switched systems," IEEE Trans. Autom. Control, vol. 54, no. 6, pp. 1430-1435, Jun. 2009.
39. F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. New York: Springer-Verlag, 2003.
40. C. Seatzu, D. Corona, A. Giua, and A. Bemporad, "Optimal control of continuous-time switched affine systems," IEEE Trans. Autom. Control, vol. 51, no. 5, pp. 726-741, May 2006. (Pubitemid 43792635)
41. C. Yfoulis and R. Shorten, "A numerical technique for stability analysis of linear switched systems," in Hybrid Systems: Comput. and Control. Philadelphia, PA: Springer-Verlag, 2004, pp. 631-645.
42. D. Stanford and J. Urbano, "Some convergence properties of matrix sets," SIAM J. Matrix Anal. Appl., vol. 15, no. 4, pp. 1132-1140, 1994.
43. W. Zhang, A. Abate, and J. Hu, "Efficient suboptimal solutions of switched LQR problems," in Proc. Amer. Control Conf., St. Louis, MO, 2009, pp. 1084-1091.
44. W. Zhang and J. Hu, "Switched LQR problem in discrete time: Theory and algorithm," Purdue Univ., West Lafayette, IN, Tech. rep. TR-ECE 09-03, 2010.
45. I. Polik and T. Terlaky, "A survey of the S-lemma," SIAM Rev., vol. 49, no. 3, pp. 371-418, 2007. (Pubitemid 47377690)
46. J. Hu, J. Shen, and W. Zhang, "A generating function approach to the stability analysis of switched linear systems," in Proc. 13th ACM Int. Conf. Hybrid Syst., Comput. and Control, Stockholm, Sweden, 2010, pp. 273-282.
47. E. Hille, Analytic Function Theory. Providence, RI: AMS Chelsea, 1959, vol. I.