We organize the input and output samples of a linear time invariant (LTI) system into two matrices, $Y$ and $G$, following a specific pattern. It is understood that a linear relationship exists between these matrices, expressed as $XG=Y$. The unknown matrix $X$ can be easily computed using the linear regression formula.
The challenge arises when there is noise present in the output measurements, which gets incorporated into the $Y$ and $G$ matrices. In such instances, how can we determine the same $X$?
Considering that we can acquire as much data from the system as needed, the dimensions of $Y$ and $G$ can be expanded as necessary.
I am a control engineering student and I don't know much about all kinds of regression methods. This is a problem that I have encountered in the practical implementation of a data driven control system.