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.
翻译:动态系统(即状态-空间形式的ODE系统)往往可以方便地描述现实世界现象。然而,如果只部分观察系统的状态,那么观察到的数量(产出)和系统投入通常可以通过更复杂的差分星位方程式(DAEs)联系起来。因此,一个自然的问题(称为可变性问题)是:给定一个差异-升位方程式(根据数据进行计算),它是否来自一个部分观测到的动态系统?一个特别的例子,即动态系统中的功能是理性的。对于单一输出变量中的单一差异-地缘方程式,Forsman已经表明,如果并且只有在相应的超表层是统一的情况下,它才被一个理性的动态方程式(DA)所实现。在这个文件中,我们研究了一个比较一般的单一投入-输出方程式案例。我们表明,如果一个理性的动态系统能够完全实现,那么,一个单一的数值方程式系统就可以用来为DAA提供更高层次的系统。我们使用了一个高级的算法。