A free response signal is generated by input shaping from the input and output signals of an unknown SISO linear system that includes delay time. The denominator of the transfer function is modeled based on this generated free response. Next, by swapping the input and output, the numerator can be modeled in a similar method. From the above, a transfer function that does not include delay time is modeled. Furthermore, the delay time is identified from the phase difference between the output of the identified transfer function and the actual output. This enables simple and highly accurate system identification for an unknown linear system.
Nishida et al. (2025) studied this question.