The switched system with possible unstable modes is investigated by employing both stabilizing and destabilizing switching instants under the time-dependent switching signal. A new piece-wise Lyapunov function is proposed to distinguish the property of the switching instants with the help of dwell time information. Then, globally exponentially stable conditions of the switched nonlinear system and the switched linear system are derived, respectively, which compensate the state divergence resulted from unstable modes and destabilizing switching instants by stabilizing switching instants. The related result is extended to the controller design of the switched linear system. Besides, for the switched linear systems with pairwise commutative, a necessary and sufficient condition is proposed based on a new time-dependent switching signal, which provides a novel way for the controller design of the switched system. Finally, simulation is presented to show the effectiveness of our results.

通过在与时间有关的开关信号下采用稳定和不稳定的开关瞬间，研究了具有可能不稳定模式的开关系统。提出了一种新的分段Lyapunov函数，以借助驻留时间信息来区分切换瞬间的属性。然后，分别推导了非线性切换系统和线性切换系统的全局指数稳定条件，该条件补偿了不稳定模式引起的状态发散，并通过稳定开关时刻来使开关时刻不稳定。相关结果扩展到开关线性系统的控制器设计。此外，对于成对换向的线性切换系统，基于新的时间相关切换信号提出了充要条件，这为交换系统的控制器设计提供了一种新颖的方法。最后，仿真表明了我们的结果的有效性。