论文标题
通过理性动力学系统实现微分 - 代数方程
On realizing differential-algebraic equations by rational dynamical systems
论文作者
论文摘要
现实现象通常可以通过动态系统（即状态空间形式的ODE系统）方便地描述。但是，如果仅部分观察系统的状态，则观察到的数量（输出）和系统的输入通常可以与更复杂的差分 - 代码方程（DAE）相关。因此，一个自然的问题（称为可实现的问题）是：给定一个差分 - 代数方程（例如，从数据拟合），是否来自部分观察到的动态系统？一个特殊情况是，动态系统中涉及的功能特别令人感兴趣。对于单个输出变量中的单个微分 - 代数方程，福斯曼表明，当且仅当相应的超曲面是Unirational时，他可以通过理性动力学系统实现，并且在第一阶情况下，他将其变成了算法。 在本文中，我们研究了单输入单输出方程的更一般情况。我们表明，如果存在理性动力学系统的实现，则可以将系统的维度等于DAE的顺序。我们为一阶DAE提供了完整的算法。我们还表明，使用文献中的几个示例可以将相同的方法用于高阶DAE。
Real-world phenomena can often be conveniently described by dynamical systems (that is, ODE systems in the state-space form). However, if one observes the state of the system only partially, the observed quantities (outputs) and the inputs of the system can typically be related by more complicated differential-algebraic equations (DAEs). Therefore, a natural question (referred to as the realizability problem) is: given a differential-algebraic equation (say, fitted from data), does it come from a partially observed dynamical system? A special case in which the functions involved in the dynamical system are rational is of particular interest. For a single differential-algebraic equation in a single output variable, Forsman has shown that it is realizable by a rational dynamical system if and only if the corresponding hypersurface is unirational, and he turned this into an algorithm in the first-order case. In this paper, we study a more general case of single-input-single-output equations. We show that if a realization by a rational dynamical system exists, the system can be taken to have the dimension equal to the order of the DAE. We provide a complete algorithm for first-order DAEs. We also show that the same approach can be used for higher-order DAEs using several examples from the literature.
