Modern modeling languages for general physical systems, such as Modelica, Amesim, or Simscape, rely on Differential Algebraic Equations (DAEs), i.e., constraints of the form $f(x^{\prime },x,u)=0$. This drastically facilitates modeling from first principles of the physics, as well as the reuse of models. In this paper, we develop the mathematical theory needed to establish the development of compilers and tools for DAE-based physical modeling languages on solid mathematical bases.

Unlike Ordinary Differential Equations (ODEs, of the form $x^{\prime }=g(x,u)$), DAEs exhibit subtle issues because of the notion of differentiation index and related latent equations—ODEs are DAEs of index zero, for which no latent equation needs to be considered. Prior to generating execution code and calling solvers, the compilation of such languages requires a nontrivial structural analysis step that reduces the differentiation index to a level acceptable by DAE solvers.

The models supported by tools of the Modelica class involve multiple modes, with mode-dependent DAE-based dynamics and state-dependent mode switching. However, multimode DAEs are much more difficult to handle than DAEs, especially because of the events of mode change. Unfortunately, the large literature devoted to the mathematical analysis of DAEs does not cover the multimode case, typically saying nothing about mode changes. This lack of foundations causes numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes.

In this paper, we develop a comprehensive mathematical approach supporting compilation and code generation for this class of languages. Its core is the structural analysis of multimode DAE systems. As a byproduct of this structural analysis, we propose sound criteria for accepting or rejecting multimode models. Our mathematical development relies on nonstandard analysis, which allows us to cast hybrid system dynamics to discrete-time dynamics with infinitesimal step size, thus providing a uniform framework for handling both continuous dynamics and mode change events.

用于一般物理系统的现代建模语言（例如Modelica，Amesim或Simscape）依赖于微分代数方程（DAE），即形式的约束 $F（X^{\prime }，X，ü）=0$。这极大地促进了从物理学的第一原理进行建模以及模型的重用。在本文中，我们开发了必要的数学理论，以建立基于DAE的物理建模语言在坚实的数学基础上的编译器和工具的开发。

不像常微分方程（ODE）的形式 $X^{\prime }=G（X，ü）$），由于微分指数和相关的潜伏方程的概念，DAE表现出细微的问题-ODE是指数为零的DAE，因此不需要考虑潜伏方程。在生成执行代码和调用求解器之前，此类语言的编译需要一个非平凡的结构分析步骤，该步骤必须将差异化索引降低到DAE求解器可接受的水平。

Modelica类工具支持的模型涉及多种模式，包括基于模式的DAE动态和基于状态的模式切换。但是，多模DAE比DAE难处理得多，尤其是由于模式改变的事件。不幸的是，致力于DAE的数学分析的大量文献并未涵盖多模式情况，通常没有提及模式变化。基础的缺乏给现有的建模工具带来了许多困难。有些模型处理得很好，另一些模型则不行，两个类之间没有明确的界限。

在本文中，我们开发了一种全面的数学方法来支持此类语言的编译和代码生成。其核心是多模DAE系统的结构分析。作为此结构分析的副产品，我们提出了接受或拒绝多模模型的合理标准。我们的数学开发依赖于非标准分析，该分析使我们能够将混合系统动力学转换为具有无限小步长的离散时间动力学，从而为处理连续动力学和模式变化事件提供了统一的框架。