# Time domain Models

Course subject(s) 06. Time domain Models

Continuous systems can be approximated well by discrete time models. Discrete time models are in fact the regression coefficients of a discrete impulse response function. Having N discrete signal values, discrete models normally require far less regression coefficients (n<N) compared to FRFs (N). Major advantage of time domain models is that in the time domain noise can be separated from signals. Compared to the frequency domain where noise is mixed with the signal that requires a posteriori averaging, in the discrete time domain noise models are estimated a priori. Immediately, this a priori knowledge of the system’s structure which is often not known beforehand. In this lecture, different time domain models are presented. Besides the models are all linear input-output models, they are not all linear in their parameters (i.e. regression coefficients). Linearity in the parameters means a linear contribution of the parameters to the model error, which is typically the difference of the modeled output and  the system’s output. Advantage of linearity in the parameters is that  the model parameters can be obtained algebraically from the input and output signals only. Linearity in the parameters depends on the chosen noise model. E.g. ARX is linear in its parameters and ARMAX is nonlinear in  its parameters. Two discrete closed loop estimators are presented (two stage and coprime factorization), both utilizing two open loop estimation steps but each in a different way.

## Time domain models 