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 continuous-time subsystems are Hurwitz stable, all discrete-time subsystems are Schur stable, and furthermore the obtained 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. A numerical example is provided to demonstrate the result.