打破密集结构：证明密集结构混合系统的稳定性
Breaking Dense Structures: Proving Stability of Densely Structured Hybrid Systems
AI总结：
本文改进了混合系统的分解技术，通过放松图结构提升效率并扩展可分解的图结构范围。
摘要：
抽象和细化在软件开发中被广泛应用。这些技术的价值在于允许处理更复杂的系统。一个关键点是能够将大系统分解为子系统，分析这些子系统并推断出更大系统的属性。随着网络物理系统变得越来越复杂，此类技术变得更具吸引力。2009年，Oehlerking和Theel提出了一种（分解）技术用于混合系统。该技术基于图，并为具有复杂离散状态空间的混合系统构造Lyapunov函数。该技术包括（1）将混合系统的底层图分解为子图，（2）为子图计算多个局部Lyapunov函数，最后（3）将局部Lyapunov函数组合成分段Lyapunov函数。Lyapunov函数可以用于多种目的，例如证明系统的稳定性或终止，或允许构造不变集，这些不变集可以用来证明安全性和安全性。在本文中，我们提出了一种改进的分解技术，该技术在应用分解技术之前放松了图结构。我们的放松显著减少了图的连接性，通过利用超密集切换。放松使分解技术在一方面更加高效，另一方面允许分解更广泛的图结构。
英文摘要：
Abstraction and refinement is widely used in software development. Such techniques are valuable since they allow to handle even more complex systems. One key point is the ability to decompose a large system into subsystems, analyze those subsystems and deduce properties of the larger system. As cyber-physical systems tend to become more and more complex, such techniques become more appealing. In 2009, Oehlerking and Theel presented a (de-)composition technique for hybrid systems. This technique is graph-based and constructs a Lyapunov function for hybrid systems having a complex discrete state space. The technique consists of (1) decomposing the underlying graph of the hybrid system into subgraphs, (2) computing multiple local Lyapunov functions for the subgraphs, and finally (3) composing the local Lyapunov functions into a piecewise Lyapunov function. A Lyapunov function can serve multiple purposes, e.g., it certifies stability or termination of a system or allows to construct invariant sets, which in turn may be used to certify safety and security. In this paper, we propose an improvement to the decomposing technique, which relaxes the graph structure before applying the decomposition technique. Our relaxation significantly reduces the connectivity of the graph by exploiting super-dense switching. The relaxation makes the decomposition technique more efficient on one hand and on the other allows to decompose a wider range of graph structures.
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