Time domain input and output measurements should be used for system identification if they are recorded simultaneously in real time and if all measurement delays are properly accounted for. The time offset between the output and the input signals is very critical as it is associated with important system characteristics such as the phase lag.
If the signals cannot be recorded simultaneously, one should consider identifying the system using frequency domain methods. As an example, one could use the original time-domain data to construct an "iddata" object. Then, the "spafdr" function could be used to create a frequency response estimate. The "spafdr" function will return an "idfrd" object which can be used to estimate a transfer function using "tfest".
The advantage of doing system identification in frequency domain is that the influence of the time offset between the signals is eliminated, and hence two separately recorded signals can be used.
Please refer to the following link for a detailed case study on system identification using recorded signals:
