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态系统，其当前模式及描述其行为的常微分方程组完全由外部数字输入信号决定，而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。此类系统的显著特征在于易于组合——这一特性得益于其纯数字接口。阈值混合系统最具前景的应用领域之一是数字集成电路：现代数字电路设计将数百万乃至数十亿个基本逻辑门（如反相器、或门、与非门）视为组合体。由于每个逻辑门本质上均由电子电路实现，且其行为受常微分方程组支配，因此阈值混合系统完美适用于从模拟世界到数字世界的严格过渡。本文证明，对于一大类阈值混合系统，从数字输入信号到数字输出信号的映射具有连续性。进一步地，我们表明在关于因果性的温和条件下，这种连续性对任意组合仍保持成立，从而保证组合系统能够忠实还原模拟现实。通过将我们的通用结果应用于近期开发的阈值混合门级模型（包括反相器等单输入单输出门以及二输入CMOS或非门），验证了这些模型的连续性。此外，我们提出了一种新型二输入或非门阈值混合模型，该模型不仅具备连续性，且与现有模型不同，能忠实模拟所有多输入开关效应。