Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time Subsystems

• Published in: International Journal of Applied Mathematics and Computer Science Vol. 17; no. 4; pp. 447 - 454
• Main Authors: Zhai, Guisheng, Xu, Xuping, Lin, Hai, Liu, Derong
• Format: Journal Article
• Language: English
• Published: Zielona Góra Versita 01.12.2007; De Gruyter Poland
• Subjects: arbitrary switching; common quadratic Lyapunov functions; dwell time scheme; exponential stability; Lie algebra; switched systems
• ISSN: 1641-876X; 2083-8492
• DOI: 10.2478/v10006-007-0036-x

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.