微分代数方程的有理动力系统实现
On realizing differential-algebraic equations by rational dynamical systems
摘要：
现实世界的现象通常可以方便地用动力系统（即状态空间形式的常微分方程系统）来描述。然而，如果一个观察系统的状态只有部分，所观察到的数量（输出）和系统的输入通常可以通过更复杂的微分代数方程（DAE）。因此，一个自然的问题（称为可实现性问题）是：给定一个微分代数方程（比如说，从数据拟合），它是否来自一个部分观测的动力系统？一个特殊的情况下，其中涉及的功能在动力系统是合理的是特别感兴趣的。对于单个输出变量的单个微分代数方程，Forsman证明了它可由有理动力系统实现当且仅当相应的超曲面是单有理的，并且他将其转化为一阶情况下的算法。本文研究了一类更一般的单输入单输出方程。我们表明，如果一个理性的动力系统的实现存在，该系统可以采取的维数等于的DAE的顺序。我们提供了一个完整的一阶DAE算法。我们还表明，同样的方法可以用于高阶DAE使用文献中的几个例子。
英文摘要：
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.
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