物理系统建模语言的数学基础
The Mathematical Foundations of Physical Systems Modeling Languages
摘要：
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f（dot{x}，x，u）=0的约束。这极大地促进了从物理学的第一原理建模和模型的重用。在本文中，我们开发的数学理论需要建立基于DAE的物理建模语言的编译器和工具的发展坚实的数学基础。与常微分方程不同，DAE表现出微妙的问题，因为微分指数和相关潜方程的概念- ODE是指数为零的DAE，不需要考虑潜方程。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重要的\emph{structural analysis}步骤，该步骤将微分索引降低到DAE求解器可接受的水平。Modelica类工具支持的模型涉及多个模式，这些模式具有基于模式依赖DAE的动态特性和状态依赖模式切换。多模式DAE比DAE困难得多。主要的困难是模式改变事件的处理。不幸的是，大量的文献致力于DAE的数学分析不包括多模情况下，通常没有说模式的变化。这种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，有些则不然，两个类之间没有明确的界限。在本文中，我们开发了一个全面的数学方法支持编译和代码生成这类语言。其核心是多模DAE系统的结构分析。作为这种结构分析的副产品，我们提出了接受或拒绝模型的合理标准。对于我们的数学发展，我们依赖于非标准分析，这使我们能够铸造混合动力系统的动态离散时间动态与无穷小的步长，从而提供了一个统一的框架来处理连续动态和模式变化事件。
英文摘要：
Modern modeling languages for general physical systems, such as Modelica, Amesim, or Simscape, rely on Differential Algebraic Equations (DAE), i.e., constraints of the form f(dot{x},x,u)=0. This drastically facilitates modeling from first principles of the physics and 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, DAE exhibit subtle issues because of the notion of differentiation index and related latent equations -- ODE are DAE 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 \emph{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. Multimode DAE are much more difficult than DAE. The main difficulty is the handling 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 well sound criteria for accepting or rejecting models. For our mathematical development, we rely on nonstandard analysis, which allows us to cast hybrid systems dynamics to discrete time dynamics with infinitesimal step size, thus providing a uniform framework for handling both continuous dynamics and mode change events.
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