Model parameter estimation of linear time-invariant systems from combined data of forced and initial condition responses

Many systems can be represented as linear time-invariant (LTI) systems in state space with ordinary differential equations (ODE). Forced responses are often used for model parameter estimation; however, some models are not uniquely identifiable from the data of forced responses, or experiments with pure forced response may not be the optimal design. It is thus meaningful to look for other types of data for model parameter estimation through redesigning experiments. In this work, we compare the influence of forced and initial condition responses on the deterministic identifiability of LTI systems in state space with ODEs as model structure. It is clearly demonstrated that one initial condition vector is equivalent to one column vector of the control matrix for constraining system eigenvectors. The combination of forced and initial condition responses can improve the identifiability of models that are not identifiable only from forced responses. Explicit formulations and an algorithm are derived to identify model parameters from the combined data of forced and initial condition responses.