In this paper, we investigate the stability and periodicity of a class of state-dependent switched systems with all unstable subsystems by means of energy analysis. We firstly transform the unstable subsystems reversibly into the form of second order mechanical systems, and then construct energy functions by calculating the sum of kinetic and potential energies of each subsystem. After that, two switching lines, derived from the lines with the largest and smallest energy drops, make the stable phase trajectory approach to the equilibrium point at the fastest speed. In addition, we explore possible dynamic behaviors of the switched system under a pair of switching line including asymptotic stability, instability and periodicity. Furthermore, based on the bisection method and nested intervals theorem, we design a state-dependent switching law, which makes the switched system periodic initiated from a stable switching law. Finally, numerical simulation examples are provided to illustrate the effectiveness and less conservativeness of the proposed method with practical significance.

在本文中，我们通过能量分析研究了一类具有所有不稳定子系统的状态相关切换系统的稳定性和周期性。我们首先将不稳定的子系统可逆地转换为二阶机械系统的形式，然后通过计算每个子系统的动能和势能之和来构造能量函数。此后，从具有最大和最小能量降的线中派生出两条开关线，以最快的速度使稳定的相轨迹接近平衡点。此外，我们探索了在一对切换线下的切换系统可能的动态行为，包括渐近稳定性，不稳定性和周期性。此外，基于对分法和嵌套区间定理，我们设计了一个状态相关的切换定律，这使交换系统从稳定的交换定律周期性地启动。最后，通过数值算例说明了该方法的有效性和保守性，具有实际意义。