Stability and £2 Gain Analysis of Switched Symmetric Systems
We study stability and £2 gain properties for a class of switched systems that are composed of a finite number of linear time-invariant symmetric subsystems. We consider both continuous-time and discrete-time switched systems. We show that when all the subsystems are (Hurwitz or Schur) stable, the switched system is exponentially stable under arbitrary switching. Furthermore, we show that when all the subsystems have £2 gain less than a positive scalar γ,the switched system has £2 gain less than the same γ under arbitrary switching. We also extend the results to switched symmetric systems with time delay in system state for continuous-time switched systems. The key idea for both stability and £2 gain analysis in this chapter is to establish a common Lyapunov function for all the subsystems in the switched system.
KeywordsLyapunov Function Linear Matrix Inequality Network Control System Symmetric System Switching Signal
Unable to display preview. Download preview PDF.
- J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” Proc. 38th IEEE ConfEvent on Decision and Control, pp. 2655–2660, Phoenix, AZ, 1999.Google Scholar
- M. A. Wicks, P. Peleties, and R. A. DeCarlo, “Construction of piecewise Lyapunov functions for stabilizing switched systems,” Proc. 33rd IEEE ConfEvent on Decision and Control, pp. 3492–3497, Orlando, FL, 1994.Google Scholar
- B. Hu, X. Xu, A. N. Michel, and P. J. Antsaklis, “Stability analysis for a class of nonlinear switched systems,” Proc. 38th IEEE ConfEvent Decision and Control, pp. 4374–4379, Phoenix, AZ, 1999.Google Scholar
- G. Zhai and K. Yasuda, “Stability analysis for a class of switched systems,” Trans. Society of Instrument and Control Engineers, vol. 36, no. 5, pp. 409–415, 2000.Google Scholar
- S. Pettersson and B. Lennartson, “LMI for stability and robustness of hybrid systems,” Proc. American Control ConfEvent, pp. 1714–1718, Albuquerque, NM, 1997.Google Scholar
- G. Zhai, “Quadratic stabilizability of discrete-time switched systems via state and output feedback,” Proc. 40th IEEE ConfEvent Decision and Control, pp. 2165–2166, Orlando, FL, 2001.Google Scholar
- J. P. Hespanha, Logic-Based Switching Algorithms in Control, Ph.D. Dissertation, Yale University, 1998.Google Scholar
- J. P. Hespanha, “Computation of £2-induced norms of switched linear systems,” Proc. 5th International Workshop of Hybrid Systems: Computation and Control, pp. 238–252, Stanford University, Stanford, CA, 2002.Google Scholar
- M. Rubensson and B. Lennartson, “Stability and robustness of hybrid systems using discrete-time Lyapunov techniques,” Proc. American Control ConfEvent, pp. 210–214, Chicago, IL, 2000.Google Scholar
- M. Ikeda, “Symmetric controllers for symmetric plants,” Proc. 3rd European Control ConfEvent, pp. 988–993, Rome, Italy, 1995.Google Scholar
- M. Ikeda, K. Miki, and G. Zhai, “H∞ controllers for symmetric systems: A theory for attitude control of large space structures,” Proc. 2001 International ConfEvent on Control,Automation and Systems, pp. 651–654, Cheju National University, Korea, 2001.Google Scholar
- T. Iwasaki, R. E. Skelton, and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1998.Google Scholar