论文标题
物理系统建模语言的数学基础
The Mathematical Foundations of Physical Systems Modeling Languages
论文作者
论文摘要
通用物理系统（例如Modelica，Amesim或Simscape）的现代建模语言依赖于差异代数方程（DAE），即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和模型重用的建模。在本文中，我们开发了建立基于DAE的物理建模语言的编译器和工具所需的数学理论。与普通的微分方程不同，DAE由于分化指数和相关潜在方程的概念而表现出微妙的问题 - ode是零索引零的dae，不需要考虑潜在方程。在生成执行代码和调用求解器之前，此类语言的汇编需要一个非平凡的\ emph {结构分析}步骤，该步骤将分化索引降低到DAE solvers可以接受的级别。 Modelica类工具支持的模型涉及多种模式，这些模式与模式依赖性DAE的动力学和与状态有关的模式切换。多模DAE比DAE难得多。主要困难是处理模式变化事件。 不幸的是，致力于大家的数学分析的大型文献并不涵盖多模案例，通常对模式变化一无所知。 缺乏基础会给现有的建模工具带来许多困难。 有些模型处理得很好，其他模型则没有，这两个类之间没有明确的边界。 在本文中，我们开发了一种全面的数学方法，该方法支持此类语言的汇编和代码生成。它的核心是多模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良好声音标准。对于我们的数学开发，我们依靠非标准分析，这使我们能够施放混合系统动力学以无限的步长离散时间动态，从而为处理连续动态和模式变更事件提供了统一的框架。
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.
