Abstract This paper presents a novel abstraction technique for analyzing Lyapunov and asymptotic stability of polyhedral switched systems. A polyhedral switched system is a hybrid system in which the continuous dynamics is specified by polyhedral differential inclusions, the invariants and guards are specified by polyhedral sets and the switching between the modes do not involve reset of variables. A finite state weighted graph abstracting the polyhedral switched system is constructed from a finite partition of the state–space, such that the satisfaction of certain graph conditions, such as the absence of cycles with product of weights on the edges greater than (or equal) to 1, implies the stability of the system. However, the graph is in general conservative and hence, the violation of the graph conditions does not imply instability. If the analysis fails to establish stability due to the conservativeness in the approximation, a counterexample (cycle with product of edge weights greater than or equal to 1) indicating a potential reason for the failure is returned. Further, a more precise approximation of the switched system can be constructed by considering a finer partition of the state–space in the construction of the finite weighted graph. We present experimental results on analyzing stability of switched systems using the above method.

中文翻译：

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基于抽象的多面体切换系统稳定性验证

摘要 本文提出了一种新的抽象技术，用于分析多面体切换系统的李雅普诺夫和渐近稳定性。多面体切换系统是一种混合系统，其中连续动力学由多面体微分包含指定，不变量和保护由多面体集指定，模式之间的切换不涉及变量的重置。抽象多面体切换系统的有限状态加权图是从状态空间的有限分区构建的，从而满足某些图条件，例如不存在边上的权重乘积大于（或等于）的循环到 1，意味着系统的稳定性。然而，该图通常是保守的，因此，违反图条件并不意味着不稳定。如果分析由于近似的保守性而未能建立稳定性，则返回一个反例（边权重的乘积大于或等于 1 的循环）指示失败的潜在原因。此外，通过在有限加权图的构造中考虑状态空间的更精细划分，可以构造更精确的切换系统近似值。我们展示了使用上述方法分析切换系统稳定性的实验结果。通过在有限加权图的构造中考虑状态空间的更精细划分，可以构造更精确的切换系统近似值。我们展示了使用上述方法分析切换系统稳定性的实验结果。通过在有限加权图的构造中考虑状态空间的更精细划分，可以构造更精确的切换系统近似值。我们展示了使用上述方法分析切换系统稳定性的实验结果。