Stability analysis of switching systems with all modes unstable based on a $\mathrm{\Phi }$-dependent max-minimum dwell time method

This paper applies stable and unstable switching instants to study switching systems with unstable modes under time-dependent switching signals. A new Lyapunov function is proposed to study the properties of switching instant using different $\mathrm{\Phi }$-dependent max-minimum dwell time methods. After that, the global exponential stability conditions for nonlinear and linear switching systems are derived and stable switching is used to compensate for unstable switching and unstable modes. The relevant results are extended to the controller design of linear switching systems. Finally, a simulation verifies the effectiveness of the results of this paper.