This paper proposes a new method for designing a state-dependent switching law to stabilize a class of switched nonlinear system with two unstable subsystems. The main idea is to convert each subsystem to a second-order mechanical system by introducing a reversible transformation, thus the summation of its kinetic and potential energies is calculated as an energy function. Then, by defining a performance index with energy function and using the variational principle, two optimal switching curves are derived from the Euler equation. By adopting a switching law designed by such switching curves, the state of the switched system can approach to the origin at the fastest speed in a sense. In addition, for the case of switched systems with linear vibration subsystems, a critical stability condition related to the stiffness and negative damping coefficients of the subsystems is obtained to make the switched system periodic. Finally, simulation results show that the proposed method can effectively solve the stability problem of switched systems, in which each subsystem does not have any stability factors.

本文提出了一种新的方法，用于设计状态相关的切换定律，以稳定具有两个不稳定子系统的一类切换非线性系统。主要思想是通过引入可逆变换将每个子系统转换为二阶机械系统，因此将其动能和势能之和计算为能量函数。然后，通过定义具有能量函数的性能指标并使用变分原理，从欧拉方程中得出两条最佳开关曲线。通过采用由这种切换曲线设计的切换定律，在某种意义上，切换系统的状态可以以最快的速度接近原点。此外，对于带有线性振动子系统的开关系统，获得与子系统的刚度和负阻尼系数有关的临界稳定性条件，以使切换系统具有周期性。最后，仿真结果表明，所提出的方法可以有效地解决每个子系统没有任何稳定性因子的切换系统的稳定性问题。