I have a doubt concerning recursive system identification. I have seen that in many occasions we have a data set $u(t)=[u(1)\, u(2) \, \dots]$. When the first couple of data $y(t)$ and $u(t)$ arrives, $\theta$ is updated (and previous data ($t<1$) for input and output is assumed to be zero). However, what I have seen is that, if the model doesn't have any delay, the initial guess for theta is overriten by the update (in the case of MATLAB it means that the first row of estimated $\theta$ is no longer $\theta_0$, but $\theta_1$).
As I'm performing continuous-time identification, I need to estimate derivatives of the signals, which requires at least two datapoints. Even though I could have proceeded in the traditional way (assuming $u(0)=0$ and $y(0)=0$ in order to be able to compute derivatives), what I have done is just waiting until I also have $u(2)$ and $y(2)$ and then use $u(1),u(2),y(1),y(2)$ in order to perform recursions and update $\theta$. In this scenario, the first row of $\theta$ is always $\theta_0$. I'm not sure if this approach is wrong (although in practice it manages to correctly estimate the parameters!) and would like to know the opinion of the community.
Note: I am Indeed identifying a continous-time model. However, in a realistic experiment the input-output data one gets IS normally sampled. Therefore, scientific littérature concerning continous-time model identificación by using sampled data is known (for example, see the Contsid toolbox). My question is about the correctness of my alternative method for the initial iteration (in practice in simulation, this alternative seems to work fine).
