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.

The stability and stabilization problems of discrete-time switched systems are studied under the so-called bipartite persistent dwell-time switching, which is proposed by relaxing some of the limitations in existing persistent dwell-time switching. This paper provides new stability criteria for discrete-time switched systems using a binary quasi-time-varying Lyapunov function. Next, the stabilizing controllers for discrete-time switched-controlled systems are designed. Finally, we give a practical example to show the effectiveness of the conclusions and less conservatism than those based on the persistent dwell-time switching.