Thresholded hybrid systems are restricted dynamical systems, where the current mode, and hence the ODE system describing its behavior, is solely determined by externally supplied digital input signals and where the only output signals are digital ones generated by comparing an internal state variable to a threshold value. An attractive feature of such systems is easy composition, which is facilitated by their purely digital interface. A particularly promising application domain of thresholded hybrid systems is digital integrated circuits: Modern digital circuit design considers them as a composition of Millions and even Billions of elementary logic gates, like inverters, GOR and Gand. Since every such logic gate is eventually implemented as an electronic circuit, however, which exhibits a behavior that is governed by some ODE system, thresholded hybrid systems are ideally suited for making the transition from the analog to the digital world rigorous. In this paper, we prove that the mapping from digital input signals to digital output signals is continuous for a large class of thresholded hybrid systems. Moreover, we show that, under some mild conditions regarding causality, this continuity also continues to hold for arbitrary compositions, which in turn guarantees that the composition faithfully captures the analog reality. By applying our generic results to some recently developed thresholded hybrid gate models, both for single-input single-output gates like inverters and for a two-input CMOS NOR gate, we show that they are continuous. Moreover, we provide a novel thresholded hybrid model for the two-input NOR gate, which is not only continuous but also, unlike the existing one, faithfully models all multi-input switching effects.

翻译：阈值混合系统是一类受限动力系统，其当前模式（即描述系统行为的常微分方程组）完全由外部输入的数字信号决定，且唯一输出信号是通过将内部状态变量与阈值比较生成的数字信号。此类系统的显著优势在于纯数字接口带来的易组合特性。阈值混合系统在数字集成电路领域具有重要应用前景：现代数字电路设计将数百万甚至数十亿个基本逻辑门（如反相器、或门、与非门）组合在一起。由于每个逻辑门最终都实现为电子电路并由常微分方程组支配其行为，阈值混合系统为从模拟域到数字域的严格转换提供了理想框架。本文证明，对于一大类阈值混合系统，数字输入信号到数字输出信号的映射具有连续性。此外，我们证明在关于因果性的温和条件下，这种连续性可推广至任意组合系统，从而确保组合系统忠实捕捉模拟现实。通过将通用结论应用于最新提出的阈值混合门模型（包括反相器等单输入单输出门及双输入CMOS或非门），我们验证了其连续性。进一步地，我们为双输入或非门提出新型阈值混合模型，该模型不仅保持连续性，且能忠实描述现有模型无法处理的多输入开关效应。