Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input-output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input-output identifiability is subtle. The goal of this note is to clarify this relation. The main results are: 1) identifiability implies input-output identifiability; 2) these notions coincide if the model does not have rational first integrals; 3) the field of input-output identifiable functions is generated by the coefficients of a "minimal" characteristic set of the corresponding differential ideal. We expect that some of these facts may be known to the experts in the area, but we are not aware of any articles in which these facts are stated precisely and rigorously proved.

中文翻译：

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参数可识别性和输入输出方程

结构参数可识别性是具有参数的微分模型的属性，允许在没有噪声的情况下从模型方程确定参数。评估这个问题的标准方法之一是通过输入-输出方程，特别是微分理想的特征集。可识别性和输入输出可识别性之间的精确关系是微妙的。本说明的目的是澄清这种关系。主要结果是：1）可识别性意味着输入输出可识别性；2) 如果模型没有有理第一积分，这些概念是一致的；3) 输入输出可识别函数域是由相应微分理想的“最小”特征集的系数生成的。