Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems
We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
- Department of Mechanical Engineering, Osaka Prefecture University, Sakai, Osaka, Japan
- Department of Electrical and Computer Engineering, Penn State Erie, Erie, PA, USA
- Department of Electrical Engineering University of Notre Dame, Notre Dame, Indiana, USA
- Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL, USA
- DeCarlo R., Branicky M.S., Pettersson S. and Lennartson B. (2000): Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, Vol. 88, No. 7, pp. 1069-1082.
- Gorbatsevich V.V., Onishchik A.L. and Vinberg E.B. (1994): Structure of Lie Groups and Lie Algebras. Berlin: Springer.
- Liberzon D. and Tempo R. (2003): Gradient algorithm for finding common Lyapunov functions. Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, USA, pp. 4782-4787.
- Liberzon D. (2003): Switching in Systems and Control. Boston: Birkhäuser.
- Liberzon D. and Morse A.S. (1999): Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, Vol. 19, No. 5, pp. 59-70.
- Liberzon D., Hespanha J.P. and Morse A.S. (1999): Stability of switched systems: A Lie-algebraic condition. Systems and Control Letters, Vol. 37, No. 3, pp. 117-122.
- Narendra K.S. and Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on Automatic Control, Vol. 39, No. 12, pp. 2469-2471.
- Samelson H. (1969): Notes on Lie Algebra. New York: Van Nostrand Reinhold.
- Zhai G. (2003): Stability and L2 gain analysis of switched symmetric systems, In: Stability and Control of Dynamical Systems with Applications, (D. Liu and P.J. Antsaklis, Eds.), Boston: Birkhäuser, pp. 131-152.
- Zhai G. (2001a): Quadratic stabilizability of discrete-time switched systems via state and output feedback. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, pp. 2165-2166.
- Zhai G., Hu B., Yasuda K. and Michel A.N. (2002a): Stability and L2 gain analysis of discrete-time switched systems. Transactions of the Institute of Systems, Control and Information Engineers, Vol. 15, No. 3, pp. 117-125.
- Zhai G., Chen X., Ikeda M. and Yasuda K. (2002b): Stability and L2 gain analysis for a class of switched symmetric systems. Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, pp. 4395-4400.
- Zhai G., Lin H., Michel A.N., and Yasuda K. (2004): Stability analysis for switched systems with continuous-time and discrete-time subsystems. Proceedings of the American Control Conference, Boston, MA, pp. 4555-4560.
- Zhai G., Xu X., Lin H., and Michel A.N. (2006): Analysis and design of switched normal systems. Nonlinear Analysis, Vol. 65, No. 12, pp. 2248-2259.
- Zhai G., Hu B., Yasuda K. and Michel A.N. (2001b): Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. International Journal of Systems Science, Vol. 32, No. 8, pp. 1055-1061.